Calculus: Limits and Continuity

Okay, here's a comprehensive lesson on Calculus: Limits and Continuity, designed for high school students (grades 9-12) with a deep dive into the concepts and applications. I've structured it to be thorough, engaging, and accessible, assuming no prior calculus knowledge.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're designing a roller coaster. You need to ensure the track is perfectly smooth so riders don't experience sudden jolts or, worse, have the coaster fly off the rails! This smoothness isn't just visual; it's a mathematical property called continuity. Before you can even think about the curves of the track, you need to understand how the coaster approaches each point – that's where limits come in. Or, perhaps you're creating a realistic animation of water flowing. To make it look believable, you need to understand how the water's position changes over infinitesimally small time intervals. Again, this involves the concept of limits. Limits and continuity are fundamental concepts that allow us to analyze how functions behave, especially when dealing with very small changes or approaching specific values. They are the foundation upon which calculus is built, enabling us to understand rates of change, areas under curves, and a whole host of real-world phenomena.

Calculus, and especially the concepts of limits and continuity, might seem abstract at first. However, these ideas are deeply intertwined with our everyday experiences. Think about driving a car. The speedometer constantly displays your speed, which is a rate of change (distance over time). To understand how that speedometer works, you need the idea of a limit. Or consider a recipe: you add ingredients in specific proportions to achieve a desired outcome. Changing those proportions even slightly can drastically alter the final product. This sensitivity to small changes is also related to the idea of continuity. We’ll explore these connections further as we delve into the lesson.

### 1.2 Why This Matters

Limits and continuity are not just abstract mathematical concepts confined to textbooks. They are the bedrock of calculus, which in turn is essential for numerous fields. Understanding limits and continuity provides the foundation for:

Real-world applications: Engineering (designing structures, analyzing fluid flow), physics (modeling motion, understanding electromagnetic fields), computer science (creating realistic simulations, developing algorithms), economics (modeling market behavior), and statistics (analyzing data).
Career connections: Engineers, physicists, computer scientists, data scientists, economists, financial analysts, actuaries, and many other professionals rely heavily on calculus. A strong grasp of limits and continuity opens doors to these rewarding and challenging careers.
Building on prior knowledge: This lesson builds upon your existing knowledge of algebra, functions, and graphing. You'll see how these concepts come together to form a more powerful tool for understanding the world.
Where this leads next: Mastering limits and continuity allows you to move on to more advanced calculus topics such as derivatives (rates of change) and integrals (areas and volumes). These, in turn, unlock even more advanced mathematical tools and applications. Beyond calculus, these concepts are crucial for fields like differential equations, real analysis, and numerical analysis.

### 1.3 Learning Journey Preview

In this lesson, we'll embark on a journey to understand limits and continuity. We will:

1. Define Limits: Explore the formal definition of a limit and how to calculate them.
2. Explore Limit Laws: Learn rules for manipulating and simplifying limits.
3. Investigate One-Sided Limits: Distinguish between approaching a value from the left and right.
4. Tackle Infinite Limits and Limits at Infinity: Examine what happens when a function grows without bound or as the input grows without bound.
5. Define Continuity: Understand the formal definition of continuity and its relationship to limits.
6. Explore Types of Discontinuities: Identify and classify different ways a function can be discontinuous.
7. Apply the Intermediate Value Theorem: Learn how to use the Intermediate Value Theorem to prove the existence of solutions to equations.
8. Apply the Squeeze Theorem: Learn how to find the limit of a function by "squeezing" it between two other functions whose limits are known.

We will use examples, analogies, and visual aids to make these concepts clear and intuitive. Each section will build upon the previous one, culminating in a solid understanding of limits and continuity.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

1. Explain the intuitive and formal definitions of a limit.
2. Calculate limits of functions using algebraic techniques, including factoring, rationalizing, and simplifying.
3. Evaluate one-sided limits and determine when a two-sided limit exists based on the behavior of one-sided limits.
4. Analyze infinite limits and limits at infinity to understand the asymptotic behavior of functions.
5. Define continuity at a point and on an interval, and verify continuity using the definition of a limit.
6. Identify and classify different types of discontinuities (removable, jump, infinite) in a function.
7. Apply the Intermediate Value Theorem to determine the existence of roots of a continuous function within a given interval.
8. Apply the Squeeze Theorem to find the limit of a function that is bounded by two other functions with known limits.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into limits and continuity, you should be comfortable with the following concepts:

Functions: Understanding what a function is, how to represent it (equation, graph, table), and function notation (e.g., f(x)).
Algebraic Manipulation: Factoring, simplifying expressions, solving equations, working with inequalities.
Graphing: Plotting points, interpreting graphs of functions, understanding the behavior of common functions (linear, quadratic, polynomial, rational).
Interval Notation: Expressing ranges of values using interval notation (e.g., [a, b], (a, b), (-∞, ∞)).
Basic Trigonometry: Understanding sine, cosine, and tangent functions (especially for later applications of limits and continuity).

If you need a refresher on any of these topics, review your algebra and precalculus textbooks or online resources like Khan Academy.

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## 4. MAIN CONTENT

### 4.1 Defining Limits: An Intuitive Introduction

Overview: The concept of a limit describes the behavior of a function near a particular point, without necessarily considering the function's value at that point. It's about what value the function is approaching.

The Core Concept: Imagine you're walking towards a door. As you get closer and closer, you're approaching the door, even if you never actually reach it. The limit of your position is the door. Similarly, the limit of a function f(x) as x approaches a value 'a' is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to 'a', but not necessarily equal to 'a'. We write this as:

``
lim (x→a) f(x) = L
`

This reads: "The limit of f(x) as x approaches a is equal to L." The key is the word "approaches." We're not interested in what f(a) is (in fact, f(a) might not even be defined!), but rather what value f(x) gets closer and closer to as x gets closer and closer to a.

Think of it like zooming in on a graph. As you zoom in closer and closer to the point x = a, the function's values should converge towards a single value, L. If they do, that's the limit. If the values bounce around or go off to infinity, the limit doesn't exist.

It's important to understand that a limit can exist even if the function is not defined at the point x = a. Also, the limit L can be different from the actual value of the function at that point, f(a). This distinction is crucial for understanding continuity later on.

Concrete Examples:

Example 1: Consider the function f(x) = (x^2 - 1) / (x - 1). This function is not defined at x = 1 because it would result in division by zero. However, we can find the limit as x approaches 1:

Setup: We want to find lim (x→1) (x^2 - 1) / (x - 1).
Process: We can factor the numerator: (x^2 - 1) = (x - 1)(x + 1). So, the function becomes f(x) = (x - 1)(x + 1) / (x - 1). For x ≠ 1, we can cancel the (x - 1) terms, leaving us with f(x) = x + 1.
Result: Now, as x gets closer and closer to 1, x + 1 gets closer and closer to 1 + 1 = 2. Therefore, lim (x→1) (x^2 - 1) / (x - 1) = 2.
Why this matters: This demonstrates that a limit can exist even when the function is not defined at the point.

Example 2: Consider the function g(x) = x + 3. We want to find lim (x→2) (x + 3).

Setup: We want to find lim (x→2) (x + 3).
Process: As x gets closer and closer to 2, x + 3 gets closer and closer to 2 + 3.
Result: Therefore, lim (x→2) (x + 3) = 5. In this case, the limit is equal to the function's value at that point (g(2) = 5).
Why this matters: This shows that sometimes the limit is simply the value of the function at the point it is approaching.

Analogies & Mental Models:

Think of it like... aiming a dart at a bullseye. The limit is the bullseye. You might not hit the bullseye exactly, but if you're a good dart player, your darts will cluster closer and closer to it.
Explain how the analogy maps to the concept: The bullseye represents the limit (L), and the darts represent the function's values (f(x)) as x approaches 'a'. The closer the darts are to the bullseye, the closer f(x) is to L as x approaches 'a'.
Where the analogy breaks down (limitations): The dart analogy doesn't perfectly capture the idea of a function not being defined at a point. In the limit concept, the function doesn't have to actually reach the limit, whereas the dart always has to hit a point.

Common Misconceptions:

❌ Students often think that the limit is the value of the function at the point.
✓ Actually, the limit is the value the function approaches as x approaches the point.
Why this confusion happens: The word "approaches" can be easily misinterpreted. It's crucial to emphasize that the limit is about the trend of the function, not necessarily its value at a single point.

Visual Description:

Imagine a graph of a function. As you trace the graph with your finger, moving closer and closer to a specific x-value (let's call it 'a'), notice what y-value your finger is approaching. That y-value is the limit of the function as x approaches 'a'. If the graph has a "hole" at x = a, the limit can still exist if the graph smoothly approaches that hole from both sides.

Practice Check:

What is the limit of f(x) = 2x + 1 as x approaches 3?

Answer: As x approaches 3, 2x + 1 approaches 2(3) + 1 = 7. Therefore, lim (x→3) (2x + 1) = 7.

Connection to Other Sections:

This section introduces the fundamental concept of a limit. The next sections will build upon this by exploring limit laws, one-sided limits, infinite limits, and continuity. Understanding this basic definition is essential for grasping the more advanced concepts.

### 4.2 The Formal (ε-δ) Definition of a Limit

Overview: While the intuitive definition of a limit is helpful for understanding the concept, the formal definition provides a rigorous mathematical framework for proving limits. It uses epsilon (ε) and delta (δ) to quantify "arbitrarily close."

The Core Concept: The formal definition of a limit states:

For every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Let's break this down:

ε (epsilon): Represents a small positive number that defines how close f(x) needs to be to the limit L. Think of it as a "tolerance" around L. |f(x) - L| < ε means that the distance between f(x) and L is less than ε.
δ (delta): Represents a small positive number that defines how close x needs to be to 'a'. Think of it as a "tolerance" around 'a'. 0 < |x - a| < δ means that x is within a distance of δ from 'a', but not equal to 'a' (the 0 < part excludes x = a).
The Definition in Words: No matter how small you make ε (the tolerance around L), you can always find a δ (the tolerance around 'a') such that whenever x is within δ of 'a' (but not equal to 'a'), f(x) is within ε of L.

In other words, you can make f(x) as close as you want to L (by choosing a small enough ε) by making x close enough to 'a' (by finding a suitable δ).

The formal definition is used to prove that a limit exists and is equal to a specific value. It's not typically used to calculate limits in practice, but it's essential for understanding the underlying mathematical rigor.

Concrete Examples:

Example 1: Prove that lim (x→2) (3x - 2) = 4 using the ε-δ definition.

Setup: We need to show that for any ε > 0, there exists a δ > 0 such that if 0 < |x - 2| < δ, then |(3x - 2) - 4| < ε.
Process: First, simplify the expression |(3x - 2) - 4| = |3x - 6| = 3|x - 2|. We want to make this less than ε. So, we have 3|x - 2| < ε. Divide both sides by 3: |x - 2| < ε/3.
Result: Now, we can choose δ = ε/3. If 0 < |x - 2| < δ = ε/3, then 3|x - 2| < 3(ε/3) = ε. Therefore, |(3x - 2) - 4| < ε.
Why this matters: We have shown that for any ε > 0, we can find a δ (namely, ε/3) that satisfies the formal definition. This proves that the limit is indeed 4.

Example 2: Prove that lim (x→1) x^2 = 1 using the ε-δ definition.

Setup: We need to show that for any ε > 0, there exists a δ > 0 such that if 0 < |x - 1| < δ, then |x^2 - 1| < ε.
Process: First, factor |x^2 - 1| = |(x - 1)(x + 1)| = |x - 1| |x + 1|. We want to bound |x + 1|. Assume that δ ≤ 1. Then |x - 1| < δ ≤ 1, which means -1 < x - 1 < 1, so 0 < x < 2. Therefore, 1 < x + 1 < 3, which means |x + 1| < 3. So, |x^2 - 1| = |x - 1| |x + 1| < 3|x - 1|.
Result: Now, we want 3|x - 1| < ε, so |x - 1| < ε/3. Choose δ = min(1, ε/3) (we take the minimum to ensure that our assumption δ ≤ 1 holds). If 0 < |x - 1| < δ, then |x^2 - 1| < 3|x - 1| < 3(ε/3) = ε.
Why this matters: We have shown that for any ε > 0, we can find a δ (namely, min(1, ε/3)) that satisfies the formal definition. This proves that the limit is indeed 1.

Analogies & Mental Models:

Think of it like... a game of "how close can you get?" ε is the target distance, and δ is how carefully you need to aim to hit the target. No matter how small the target (ε), you can always aim carefully enough (find a δ) to hit it.
Explain how the analogy maps to the concept: ε represents the desired accuracy of f(x) to L, and δ represents the required accuracy of x to 'a' to achieve that desired accuracy.
Where the analogy breaks down (limitations): The game analogy doesn't capture the "for all ε" aspect of the definition. It's not enough to hit the target once; you need to be able to hit it no matter how small it gets.

Common Misconceptions:

❌ Students often think that ε and δ are fixed numbers.
✓ Actually, ε is any positive number, and δ depends on ε. You need to find a δ for every ε.
Why this confusion happens: The "for every ε" part of the definition is often overlooked.

Visual Description:

Draw a graph of a function f(x). Draw a horizontal band of width 2ε around the line y = L (the limit). The ε-δ definition says that you can always find a vertical band of width 2δ around the line x = a such that whenever x is inside the vertical band (but not equal to a), f(x) is inside the horizontal band.

Practice Check:

Explain in your own words what the ε-δ definition of a limit means.

Answer: The ε-δ definition of a limit says that we can make the function's values as close as we want to the limit L by making the input x close enough to the value 'a'. No matter how small a "tolerance" we set around L (ε), we can always find a "tolerance" around 'a' (δ) that guarantees the function's values stay within that tolerance around L.

Connection to Other Sections:

This section provides the formal foundation for the intuitive understanding of limits from the previous section. While you won't be using the ε-δ definition to calculate limits in most cases, understanding it is crucial for a deep understanding of calculus. It also provides the necessary rigor for proving the limit laws in the next section.

### 4.3 Limit Laws

Overview: Limit laws are a set of rules that allow us to calculate limits of more complex functions by breaking them down into simpler components.

The Core Concept: Limit laws provide a shortcut for calculating limits without having to resort to the formal ε-δ definition every time. They state how limits interact with basic arithmetic operations and common functions. Here are some of the most important limit laws:

1. Limit of a Constant: lim (x→a) c = c (The limit of a constant is the constant itself).
2. Limit of x: lim (x→a) x = a (The limit of x as x approaches a is a).
3. Limit of a Sum/Difference: lim (x→a) [f(x) ± g(x)] = lim (x→a) f(x) ± lim (x→a) g(x) (The limit of a sum or difference is the sum or difference of the limits, provided the individual limits exist).
4. Limit of a Constant Multiple: lim (x→a) [c f(x)] = c lim (x→a) f(x) (The limit of a constant times a function is the constant times the limit of the function, provided the limit exists).
5. Limit of a Product: lim (x→a) [f(x) g(x)] = lim (x→a) f(x) lim (x→a) g(x) (The limit of a product is the product of the limits, provided the individual limits exist).
6. Limit of a Quotient: lim (x→a) [f(x) / g(x)] = [lim (x→a) f(x)] / [lim (x→a) g(x)], provided lim (x→a) g(x) ≠ 0 (The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero).
7. Limit of a Power: lim (x→a) [f(x)]^n = [lim (x→a) f(x)]^n (The limit of a function raised to a power is the limit of the function raised to that power, provided the limit exists).
8. Limit of a Root: lim (x→a) √[n]{f(x)} = √[n]{lim (x→a) f(x)} (The limit of the nth root of a function is the nth root of the limit of the function, provided the limit exists and the root is defined).
9. Limit of a Polynomial: If p(x) is a polynomial, then lim (x→a) p(x) = p(a).
10. Limit of a Rational Function: If r(x) is a rational function and 'a' is in the domain of r(x), then lim (x→a) r(x) = r(a).

Concrete Examples:

Example 1: Calculate lim (x→3) (x^2 + 2x - 1) using limit laws.

Setup: We want to find lim (x→3) (x^2 + 2x - 1).
Process: Using the limit of a sum/difference law, we can break this down into: lim (x→3) x^2 + lim (x→3) 2x - lim (x→3) 1. Using the limit of a power law, lim (x→3) x^2 = (lim (x→3) x)^2 = 3^2 = 9. Using the limit of a constant multiple law, lim (x→3) 2x = 2 lim (x→3) x = 2 3 = 6. Using the limit of a constant law, lim (x→3) 1 = 1.
Result: Therefore, lim (x→3) (x^2 + 2x - 1) = 9 + 6 - 1 = 14.
Why this matters: This demonstrates how to use multiple limit laws to simplify a complex limit calculation.

Example 2: Calculate lim (x→2) (x^3 + 4) / (x - 1) using limit laws.

Setup: We want to find lim (x→2) (x^3 + 4) / (x - 1).
Process: Using the limit of a quotient law, we can break this down into: [lim (x→2) (x^3 + 4)] / [lim (x→2) (x - 1)]. Using the limit of a sum law, lim (x→2) (x^3 + 4) = lim (x→2) x^3 + lim (x→2) 4 = 2^3 + 4 = 8 + 4 = 12. Similarly, lim (x→2) (x - 1) = lim (x→2) x - lim (x→2) 1 = 2 - 1 = 1.
Result: Therefore, lim (x→2) (x^3 + 4) / (x - 1) = 12 / 1 = 12.
Why this matters: This shows how to apply the limit laws to rational functions (functions that are ratios of polynomials).

Analogies & Mental Models:

Think of it like... a recipe. The limit laws are like the individual ingredients and the instructions on how to combine them. You can use them to "cook up" the limit of a complex function from the limits of simpler functions.
Explain how the analogy maps to the concept: Each limit law is like a specific instruction on how to handle a particular operation (addition, multiplication, etc.).
Where the analogy breaks down (limitations): The recipe analogy doesn't capture the "provided the limits exist" condition. If one of your ingredients is missing (a limit doesn't exist), you can't complete the recipe (calculate the limit).

Common Misconceptions:

❌ Students often forget the condition that the individual limits must exist for the limit laws to apply.
✓ Actually, if one of the limits doesn't exist, you can't use the limit laws to break down the problem. You need to use other techniques.
Why this confusion happens: The "provided the limits exist" condition is easy to overlook.

Visual Description:

Imagine a complex function as a building made of LEGO bricks. The limit laws allow you to take apart the building (the complex function) into individual bricks (simpler functions) and calculate the limit of each brick separately. Then, you can combine the results (the limits of the bricks) to find the limit of the whole building (the complex function).

Practice Check:

Calculate lim (x→4) (√x + 5x) using limit laws.

Answer: Using the limit of a sum law, lim (x→4) (√x + 5x) = lim (x→4) √x + lim (x→4) 5x. Using the limit of a root law, lim (x→4) √x = √(lim (x→4) x) = √4 = 2. Using the limit of a constant multiple law, lim (x→4) 5x = 5 lim (x→4) x = 5 4 = 20. Therefore, lim (x→4) (√x + 5x) = 2 + 20 = 22.

Connection to Other Sections:

This section provides the tools for calculating limits efficiently. The next sections will explore more challenging types of limits, such as one-sided limits and infinite limits, where the limit laws may need to be applied carefully.

### 4.4 One-Sided Limits

Overview: One-sided limits consider the behavior of a function as x approaches a value from either the left or the right.

The Core Concept: Sometimes, the behavior of a function as x approaches a value 'a' is different depending on whether x is approaching from values less than 'a' (the left) or values greater than 'a' (the right). This leads to the concept of one-sided limits:

Left-Hand Limit: The limit of f(x) as x approaches 'a' from the left (values less than 'a') is denoted as:

`
lim (x→a⁻) f(x) = L
`

This means that f(x) gets arbitrarily close to L as x gets arbitrarily close to 'a' while remaining less than 'a'.

Right-Hand Limit: The limit of f(x) as x approaches 'a' from the right (values greater than 'a') is denoted as:

`
lim (x→a⁺) f(x) = L
`

This means that f(x) gets arbitrarily close to L as x gets arbitrarily close to 'a' while remaining greater than 'a'.

For a two-sided limit to exist (i.e., lim (x→a) f(x) = L), both the left-hand limit and the right-hand limit must exist and be equal:

`
lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = L
`

If the left-hand limit and the right-hand limit are different, then the two-sided limit does not exist.

One-sided limits are particularly useful for analyzing functions with piecewise definitions or functions that have jumps or breaks in their graphs.

Concrete Examples:

Example 1: Consider the piecewise function:

`
f(x) = { x^2, if x < 1
{ 2x, if x ≥ 1
``

Find the left-hand limit and the right-hand limit as x approaches 1.

Setup: We need to find lim (x→1⁻) f(x) and lim (x→1⁺) f(x).
Process: For the left-hand limit, we use the definition of f(x) for x < 1, which is f(x) = x^2. So, lim (x→1⁻) f(x) = lim (x→1⁻) x^2 = 1^2 = 1. For the right-hand limit, we use the definition of f(x) for x ≥ 1, which is f(x) = 2x. So, lim (x→1⁺) f(x) = lim (x→1⁺) 2x = 2 1 = 2.
Result: lim (x→1⁻) f(x) = 1 and lim (x→1⁺) f(x) = 2. Since the left-hand limit and the right-hand limit are not equal, the two-sided limit lim (x→1) f(x) does not exist.
Why this matters: This demonstrates how one-sided limits can be used to determine whether a two-sided limit exists for a piecewise function.

Example 2: Consider the function f(x) = |x| / x.

Setup: We need to find lim (x→0⁻) f(x) and lim (x→0⁺) f(x).
Process: For x < 0, |x| = -x, so f(x) = -x / x = -1. Therefore, lim (x→0⁻) f(x) = lim (x→0⁻) -1 = -1. For x > 0, |x| = x, so f(x) = x / x = 1. Therefore, lim (x→0⁺) f(x) = lim (x→0⁺) 1 = 1.
Result: lim (x→0⁻) f(x) = -1 and lim (x→0⁺) f(x) = 1. Since the left-hand limit and the right-hand limit are not equal, the two-sided limit lim (x→0) f(x) does not exist.
Why this matters: This example shows how one-sided limits can be used to analyze functions with absolute values.

Analogies & Mental Models:

Think of it like... approaching a mountain pass. You can approach the pass from the east or from the west. If the elevation at the pass is different depending on which direction you approach from, then there's no single "elevation" at the pass.
Explain how the analogy maps to the concept: The mountain pass represents the value 'a', and the elevation represents the function's value. The east and west approaches represent the right-hand and left-hand limits, respectively.
Where the analogy breaks down (limitations): The mountain pass analogy doesn't perfectly capture the idea of a function being undefined at a point.

Common Misconceptions:

❌ Students often assume that if a function is defined at a point, then the two-sided limit must exist and be equal to the function's value at that point.
✓ Actually, the function can be defined at the point, but the one-sided limits may not exist or may not be equal to each other, in which case the two-sided limit does not exist.
Why this confusion happens: The relationship between the function's value at a point and the limit at that point can be subtle.

Visual Description:

Draw a graph of a piecewise function with a jump at x = a. As you trace the graph with your finger approaching x = a from the left, you'll approach one y-value. As you trace the graph approaching x = a from the right, you'll approach a different y-value. This visually demonstrates the concept of one-sided limits and why the two-sided limit doesn't exist.

Practice Check:

For the function f(x) = { x + 1, if x < 2; 3, if x = 2; x^2 - 1, if x > 2 }, find the left-hand limit, the right-hand limit, and the two-sided limit as x approaches 2 (if they exist).

Answer: lim (x→2⁻) f(x) = lim (x→2⁻) (x + 1) = 2 + 1 = 3. lim (x→2⁺) f(x) = lim (x→2⁺) (x^2 - 1) = 2^2 - 1 = 3. Since the left-hand limit and the right-hand limit are both equal to 3, the two-sided limit exists and lim (x→2) f(x) = 3. Note that f(2) = 3, but this is not what determines the limit.

Connection to Other Sections:

This section introduces the concept of one-sided limits, which is crucial for understanding continuity (the next major topic). A function is continuous at a point only if the two-sided limit exists and is equal to the function's value at that point. One-sided limits are also important for analyzing functions with discontinuities, which we will discuss later.

### 4.5 Infinite Limits and Limits at Infinity

Overview: This section explores what happens when a function's value grows without bound (infinite limits) or as the input variable grows without bound (limits at infinity).

The Core Concept:

Infinite Limits:

Okay, here is a comprehensive lesson on Limits and Continuity in Calculus, designed to be exceptionally detailed and suitable for high school students (grades 9-12) with a focus on deeper analysis and applications. I have aimed for clarity, depth, and engagement, keeping in mind the critical requirements you outlined.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're designing a roller coaster. You want it to be thrilling, but also safe. A key part of the design involves making sure the track is continuous – no sudden jumps or breaks that could send riders flying. Calculus, specifically the concepts of limits and continuity, provides the mathematical tools to analyze and ensure the smoothness of that roller coaster track (or the smoothness of any curve, for that matter).

Think about a video game character moving across the screen. Their position changes over time. Calculus helps us understand the rate at which their position changes (their velocity). But what if the character teleports momentarily? That's a discontinuity in their motion, and understanding limits helps us analyze what happens near that teleport, even if we can't define the character's velocity at the exact moment of teleportation. These concepts are not just abstract math; they describe how things change in the world around us.

### 1.2 Why This Matters

Understanding limits and continuity is fundamental to calculus and many other areas of mathematics, science, and engineering. These concepts are the building blocks upon which we define derivatives (rates of change) and integrals (areas and accumulations). Without them, we can't understand how a car accelerates, how a population grows, how a chemical reaction proceeds, or how a computer algorithm converges.

This knowledge is crucial for students pursuing careers in:

Engineering: Designing structures, circuits, and systems that behave predictably.
Computer Science: Developing algorithms, simulations, and graphics.
Physics: Modeling motion, forces, and energy.
Economics: Analyzing market trends and financial models.
Data Science: Understanding and predicting data patterns.

This lesson builds upon your existing knowledge of functions, graphs, and algebra. It prepares you for more advanced topics in calculus, such as differentiation, integration, and differential equations. These later topics are essential for understanding complex systems and solving real-world problems.

### 1.3 Learning Journey Preview

In this lesson, we'll embark on a journey to understand the following:

1. What is a Limit? We'll explore the intuitive idea of a limit and how to determine it graphically and numerically.
2. Formal Definition of a Limit: We'll dive into the precise epsilon-delta definition, providing a rigorous foundation.
3. Limit Laws: We'll learn rules for calculating limits of combinations of functions.
4. Limits at Infinity: We'll examine the behavior of functions as the input grows without bound.
5. Continuity: We'll define continuity and explore different types of discontinuities.
6. Intermediate Value Theorem: We'll learn about a powerful theorem that guarantees the existence of solutions to equations.
7. Applications of Limits and Continuity: We'll explore real-world examples of how these concepts are used.
8. Practice Problems: We'll work through a variety of problems to solidify your understanding.

Each concept builds on the previous one. We’ll start with the intuitive idea of a limit and gradually move towards a more formal and rigorous understanding. We will use examples, analogies, and visualizations to make the concepts clear and accessible.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

1. Explain the intuitive meaning of a limit and determine the limit of a function graphically and numerically.
2. State the formal epsilon-delta definition of a limit and apply it to prove the existence of limits for simple functions.
3. Apply limit laws to calculate the limits of sums, differences, products, quotients, and compositions of functions.
4. Evaluate limits at infinity and identify horizontal asymptotes of functions.
5. Define continuity at a point and on an interval, and classify different types of discontinuities (removable, jump, infinite).
6. State the Intermediate Value Theorem and apply it to determine the existence of roots of equations.
7. Model real-world scenarios using functions and analyze their behavior using limits and continuity.
8. Solve a variety of problems involving limits and continuity, demonstrating a strong understanding of the concepts and their applications.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into limits and continuity, you should be familiar with the following concepts:

Functions: Understanding what a function is, how to represent it (algebraically, graphically, numerically), and key concepts like domain and range.
Graphs of Functions: Ability to sketch and interpret graphs of common functions (linear, quadratic, polynomial, rational, trigonometric, exponential, logarithmic).
Algebraic Manipulation: Proficiency in simplifying expressions, solving equations, and working with inequalities.
Coordinate Geometry: Familiarity with the Cartesian coordinate system and the equation of a line.
Basic Trigonometry: Knowledge of trigonometric functions (sine, cosine, tangent) and their properties.

Quick Review:

Function: A rule that assigns to each input (x) exactly one output (y).
Domain: The set of all possible input values (x) for a function.
Range: The set of all possible output values (y) for a function.
Graph: A visual representation of a function, plotting input-output pairs (x, y) on a coordinate plane.

If you need to review any of these topics, consult your algebra or pre-calculus textbook, or online resources like Khan Academy.

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## 4. MAIN CONTENT

### 4.1 What is a Limit? - An Intuitive Introduction

Overview: The concept of a limit is fundamental to calculus. It describes the behavior of a function as its input approaches a particular value. We are interested in what value the function approaches, not necessarily the value at the point.

The Core Concept: Imagine a function, f(x). We want to know what happens to the output of the function, f(x), as the input, x, gets closer and closer to a specific value, say a. The limit is the value that f(x) "tends towards" or "approaches" as x approaches a. It's crucial to understand that the limit doesn't necessarily equal f(a). In fact, f(a) might not even be defined! We are interested in the behavior of the function near a, not at a.

Think of it like walking towards a door. The limit is the position of the door; you are approaching the door. You may or may not reach the door (f(a) may or may not be defined), but the limit describes where you are heading.

The notation for a limit is:

lim (x→a) f(x) = L

This reads: "The limit of f(x) as x approaches a is equal to L."

x→a means "x approaches a" (but is not necessarily equal to a).
f(x) is the function we are analyzing.
L is the limit – the value f(x) approaches.

Concrete Examples:

Example 1: Consider the function f(x) = x + 2. Let's find the limit as x approaches 3.

Setup: We want to find lim (x→3) (x + 2).
Process: As x gets closer and closer to 3 (e.g., 2.9, 2.99, 2.999, or 3.1, 3.01, 3.001), the value of x + 2 gets closer and closer to 5 (e.g., 4.9, 4.99, 4.999, or 5.1, 5.01, 5.001).
Result: Therefore, lim (x→3) (x + 2) = 5. In this simple case, the limit is equal to f(3), but this is not always the case.
Why this matters: This simple example illustrates the fundamental idea of approaching a value.

Example 2: Consider the function g(x) = (x² - 1) / (x - 1). Let's find the limit as x approaches 1.

Setup: We want to find lim (x→1) (x² - 1) / (x - 1).
Process: Notice that g(1) is undefined because we would be dividing by zero. However, we can simplify the expression: (x² - 1) / (x - 1) = (x + 1)(x - 1) / (x - 1) = x + 1 (for x ≠ 1). As x gets closer and closer to 1 (but is not equal to 1), the value of x + 1 gets closer and closer to 2.
Result: Therefore, lim (x→1) (x² - 1) / (x - 1) = 2. Here, the limit exists even though the function is not defined at x = 1.
Why this matters: This example shows that the limit can exist even if the function is undefined at the point it is approaching. This is a crucial concept.

Analogies & Mental Models:

Think of it like a GPS: You're driving towards a destination. The limit is the destination itself. Your actual position is f(x), and x is the time you've been driving. Even if you take a detour (a discontinuity), the GPS still knows where you're heading. The limit is the destination, regardless of the path you take to get there.
Where the analogy breaks down: A GPS can recalculate if you deviate too far. Limits don't "recalculate"; they describe the behavior as you approach a specific point.

Common Misconceptions:

❌ Students often think that the limit must equal the function's value at the point it's approaching.
✓ Actually, the limit describes the function's behavior near the point, not necessarily at the point. The function may not even be defined at that point.
Why this confusion happens: Because in many simple cases, the limit does equal the function's value. It's important to emphasize examples where this is not the case.

Visual Description:

Imagine the graph of a function. As you trace the graph from the left and the right, getting closer and closer to x = a, the y-values on the graph approach a certain y-value, L. This L is the limit. Visually, it's the "height" the function is heading towards as you get close to x = a. If the graph has a hole at x = a, the limit is the y-value of where the hole would be.

Practice Check:

What is the limit of f(x) = 2x - 1 as x approaches 4? (Answer: 7. The function approaches 7 as x approaches 4)

Connection to Other Sections: This section provides the intuitive foundation for understanding limits. The next section will delve into the formal definition, providing a more rigorous framework.

### 4.2 Formal Definition of a Limit (Epsilon-Delta)

Overview: While the intuitive idea of a limit is helpful, we need a precise, mathematical definition to prove limits and avoid ambiguity. This is where the epsilon-delta definition comes in.

The Core Concept: The epsilon-delta definition of a limit formalizes the idea of "getting arbitrarily close." It states:

For every ε > 0 (epsilon, a small positive number), there exists a δ > 0 (delta, another small positive number) such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Let's break this down:

ε (Epsilon): Represents how close we want the function's output, f(x), to be to the limit, L. Think of it as a "tolerance" around L. |f(x) - L| < ε means that f(x) is within a distance of ε from L.
δ (Delta): Represents how close x needs to be to a to ensure that f(x) is within the ε tolerance of L. Think of it as a "neighborhood" around a. 0 < |x - a| < δ means that x is within a distance of δ from a, but x is not equal to a. (The 0 < part is important because the limit doesn't care about what happens at x=a).
"For every ε > 0...": This means the definition must hold for any positive value of ε, no matter how small. We can make the tolerance around L as tight as we want.
"...there exists a δ > 0...": This means that for any chosen ε, we can find a δ that satisfies the condition. The value of δ will usually depend on ε.

In simpler terms: No matter how small you make the target around the limit (ε), you can always find a region around a (δ) such that if x is in that region (but not equal to a), then f(x) will be in the target around the limit.

Concrete Examples:

Example 1: Prove that lim (x→2) (3x - 2) = 4 using the epsilon-delta definition.

Setup: We need to show that for any ε > 0, there exists a δ > 0 such that if 0 < |x - 2| < δ, then |(3x - 2) - 4| < ε.
Process:
1. Start with the inequality |(3x - 2) - 4| < ε.
2. Simplify: |3x - 6| < ε.
3. Factor out a 3: 3|x - 2| < ε.
4. Divide by 3: |x - 2| < ε/3.
5. Now, we can choose δ = ε/3.
Result: If 0 < |x - 2| < δ = ε/3, then |(3x - 2) - 4| = 3|x - 2| < 3(ε/3) = ε. This proves the limit.
Why this matters: This example demonstrates the process of finding a δ that works for a given ε.

Example 2: Prove that lim (x→1) x² = 1 using the epsilon-delta definition.

Setup: We need to show that for any ε > 0, there exists a δ > 0 such that if 0 < |x - 1| < δ, then |x² - 1| < ε.
Process:
1. Start with the inequality |x² - 1| < ε.
2. Factor: |(x - 1)(x + 1)| < ε.
3. |x - 1| |x + 1| < ε.
4. We need to bound |x + 1|. Since x is approaching 1, we can assume that x is close to 1. Let's assume |x - 1| < 1. This implies -1 < x - 1 < 1, which means 0 < x < 2, and therefore 1 < x + 1 < 3. So, |x + 1| < 3.
5. Now we have |x - 1| |x + 1| < 3|x - 1| < ε.
6. Divide by 3: |x - 1| < ε/3.
7. Choose δ = min(1, ε/3). We take the minimum to ensure that both |x - 1| < 1 and |x - 1| < ε/3 are satisfied.
Result: If 0 < |x - 1| < δ = min(1, ε/3), then |x² - 1| = |x - 1| |x + 1| < (ε/3) 3 = ε. This proves the limit.
Why this matters: This example shows a slightly more complex proof, where we need to bound part of the expression to find a suitable δ.

Analogies & Mental Models:

Think of it like a game: You want to throw a dart and hit a target. L is the bullseye. ε is the size of the target you need to hit. δ is how accurately you need to aim your dart to hit the target. The epsilon-delta definition says that no matter how small you make the target (ε), you can always find a way to aim accurately enough (δ) to hit it.
Where the analogy breaks down: In the limit definition, x can never actually equal a. It only gets arbitrarily close.

Common Misconceptions:

❌ Students often struggle with the order of quantifiers ("for every" and "there exists").
✓ Understand that you choose ε first, and then you must find a δ that depends on your choice of ε.
Why this confusion happens: The logical structure of the definition can be difficult to grasp. Emphasize the "game" analogy to clarify the roles of ε and δ.

Visual Description:

Imagine a graph of f(x). Draw a horizontal band of width 2ε around the line y = L. The epsilon-delta definition says that you can always find a vertical band of width 2δ around the line x = a such that whenever x is inside the vertical band (but not equal to a), the corresponding f(x) value is inside the horizontal band.

Practice Check:

Explain in your own words what the epsilon-delta definition of a limit means.

Connection to Other Sections: This section provides the rigorous foundation for understanding limits. The next section will introduce limit laws, which provide tools for calculating limits more easily.

### 4.3 Limit Laws

Overview: Calculating limits directly from the epsilon-delta definition can be tedious. Fortunately, we have limit laws that allow us to calculate limits of combinations of functions more easily.

The Core Concept: Limit laws are a set of theorems that state how to find the limit of combinations of functions, assuming the individual limits exist. Here are some of the most important limit laws:

1. Limit of a Constant: lim (x→a) c = c (The limit of a constant is the constant itself).
2. Limit of x: lim (x→a) x = a
3. Sum/Difference Law: lim (x→a) [f(x) ± g(x)] = lim (x→a) f(x) ± lim (x→a) g(x) (The limit of a sum or difference is the sum or difference of the limits).
4. Constant Multiple Law: lim (x→a) [c f(x)] = c lim (x→a) f(x) (The limit of a constant times a function is the constant times the limit of the function).
5. Product Law: lim (x→a) [f(x) g(x)] = lim (x→a) f(x) lim (x→a) g(x) (The limit of a product is the product of the limits).
6. Quotient Law: lim (x→a) [f(x) / g(x)] = [lim (x→a) f(x)] / [lim (x→a) g(x)] (provided lim (x→a) g(x) ≠ 0) (The limit of a quotient is the quotient of the limits, as long as the limit of the denominator is not zero).
7. Power Law: lim (x→a) [f(x)]^n = [lim (x→a) f(x)]^n (for any positive integer n).
8. Root Law: lim (x→a) √[n]{f(x)} = √[n]{lim (x→a) f(x)} (provided lim (x→a) f(x) > 0 if n is even).

Concrete Examples:

Example 1: Find lim (x→2) (x² + 3x - 1).

Setup: We want to find the limit of a polynomial function.
Process: Using the limit laws:
lim (x→2) (x² + 3x - 1) = lim (x→2) x² + lim (x→2) 3x - lim (x→2) 1 (Sum/Difference Law)
= [lim (x→2) x]² + 3 lim (x→2) x - 1 (Power Law, Constant Multiple Law, Limit of a Constant)
= (2)² + 3 2 - 1 (Limit of x)
= 4 + 6 - 1 = 9
Result: lim (x→2) (x² + 3x - 1) = 9.
Why this matters: This example shows how to apply multiple limit laws to simplify the calculation.

Example 2: Find lim (x→3) (x² - 9) / (x - 3).

Setup: We want to find the limit of a rational function.
Process: Directly substituting x = 3 would result in division by zero. So, we first simplify the expression:
(x² - 9) / (x - 3) = (x + 3)(x - 3) / (x - 3) = x + 3 (for x ≠ 3).
Now, using the limit laws:
lim (x→3) (x² - 9) / (x - 3) = lim (x→3) (x + 3) = lim (x→3) x + lim (x→3) 3 (Sum Law)
= 3 + 3 (Limit of x, Limit of a Constant)
= 6
Result: lim (x→3) (x² - 9) / (x - 3) = 6.
Why this matters: This example shows how to simplify an expression before applying limit laws to avoid indeterminate forms.

Analogies & Mental Models:

Think of it like cooking: The limit laws are like recipes. They tell you how to combine ingredients (functions) to create a dish (a new function) and how the final dish will taste (the limit).
Where the analogy breaks down: Limit laws have specific conditions (e.g., the denominator cannot have a limit of zero). Recipes are more flexible.

Common Misconceptions:

❌ Students often try to apply the quotient law even when the limit of the denominator is zero.
✓ Remember that the quotient law only applies if the limit of the denominator is not zero.
Why this confusion happens: It's easy to forget the condition on the quotient law.

Visual Description:

The limit laws can be visualized by considering the graphs of the functions involved. For example, the sum law states that the limit of the sum of two functions is the sum of their limits. This can be visualized by adding the y-values of the two functions as x approaches a.

Practice Check:

State the quotient law and explain when it can be applied.

Connection to Other Sections: This section provides the tools for calculating limits efficiently. The next section will explore limits at infinity, which describe the behavior of functions as x becomes very large.

### 4.4 Limits at Infinity

Overview: Limits at infinity describe the behavior of a function as the input, x, grows without bound (approaches positive or negative infinity).

The Core Concept: Instead of approaching a specific number, x is growing infinitely large (or infinitely small). We want to know what happens to the output of the function, f(x), as x approaches positive infinity (x→∞) or negative infinity (x→-∞).

lim (x→∞) f(x) = L means that as x gets larger and larger, f(x) gets closer and closer to L.
lim (x→-∞) f(x) = L means that as x gets smaller and smaller (more and more negative), f(x) gets closer and closer to L.

If lim (x→∞) f(x) = L or lim (x→-∞) f(x) = L, then the line y = L is a horizontal asymptote of the function f(x). A horizontal asymptote represents a horizontal line that the function approaches as x goes to infinity or negative infinity.

Key techniques for evaluating limits at infinity:

1. Rational Functions: Divide the numerator and denominator by the highest power of x in the denominator. This simplifies the expression and allows you to evaluate the limit.
2. Polynomial Functions: If the degree of the numerator is greater than the degree of the denominator, the limit is either ∞ or -∞. If the degree of the denominator is greater than the degree of the numerator, the limit is 0.
3. Exponential and Logarithmic Functions: Understand the behavior of these functions as x approaches infinity. Exponential functions with a base greater than 1 grow without bound, while logarithmic functions grow very slowly.

Concrete Examples:

Example 1: Find lim (x→∞) (2x² + 3x - 1) / (x² + 5).

Setup: We want to find the limit of a rational function as x approaches infinity.
Process: Divide both the numerator and denominator by x² (the highest power of x in the denominator):
lim (x→∞) (2x² + 3x - 1) / (x² + 5) = lim (x→∞) (2 + 3/x - 1/x²) / (1 + 5/x²)
As x approaches infinity, 3/x, 1/x², and 5/x² all approach 0.
Therefore, the limit becomes (2 + 0 - 0) / (1 + 0) = 2/1 = 2.
Result: lim (x→∞) (2x² + 3x - 1) / (x² + 5) = 2. The line y = 2 is a horizontal asymptote.
Why this matters: This example shows how dividing by the highest power of x allows us to simplify the expression and evaluate the limit.

Example 2: Find lim (x→-∞) e^x.

Setup: We want to find the limit of an exponential function as x approaches negative infinity.
Process: As x becomes more and more negative, e^x gets closer and closer to 0.
Result: lim (x→-∞) e^x = 0. The line y = 0 is a horizontal asymptote.
Why this matters: Understanding the behavior of exponential functions is crucial for analyzing growth and decay processes.

Analogies & Mental Models:

Think of it like a race: Two runners are running towards infinity. The function with the higher degree is the faster runner. If the numerator has a higher degree, it wins the race (the limit is infinity). If the denominator has a higher degree, it wins the race (the limit is zero). If they have the same degree, it's a tie, and the limit is the ratio of their leading coefficients.
Where the analogy breaks down: This analogy only works for polynomial and rational functions. Exponential and logarithmic functions have different behaviors.

Common Misconceptions:

❌ Students often think that a function can never cross its horizontal asymptote.
✓ A function can cross its horizontal asymptote. The horizontal asymptote describes the function's end behavior, not its behavior near the origin.
Why this confusion happens: The term "asymptote" implies that the function can never touch it.

Visual Description:

Imagine the graph of a function. As you move further and further to the right (towards positive infinity) or to the left (towards negative infinity), the graph gets closer and closer to a horizontal line. This horizontal line is the horizontal asymptote.

Practice Check:

What is a horizontal asymptote, and how is it related to limits at infinity?

Connection to Other Sections: This section explores the behavior of functions as x becomes very large. The next section will introduce the concept of continuity, which describes functions that have no "breaks" or "jumps" in their graphs.

### 4.5 Continuity

Overview: Continuity is a fundamental concept in calculus that describes functions that have no "gaps," "jumps," or "holes" in their graphs.

The Core Concept: Intuitively, a continuous function is one that you can draw without lifting your pen from the paper. More formally, a function f(x) is continuous at a point x = a if the following three conditions are met:

1. f(a) is defined (the function exists at x = a).
2. lim (x→a) f(x) exists (the limit of the function as x approaches a exists).
3. lim (x→a) f(x) = f(a) (the limit of the function as x approaches a is equal to the function's value at x = a).

If any of these three conditions are not met, the function is said to be discontinuous at x = a.

There are three main types of discontinuities:

1. Removable Discontinuity: The limit exists, but the function is either undefined at the point or the limit does not equal the function value. This type of discontinuity can be "removed" by redefining the function at that point. (e.g., g(x) = (x² - 1) / (x - 1) at x = 1)
2. Jump Discontinuity: The limit from the left and the limit from the right both exist, but they are not equal. The function "jumps" from one value to another at the point of discontinuity.
3. Infinite Discontinuity: The function approaches infinity (or negative infinity) as x approaches a. This often occurs when there is a vertical asymptote at x = a.

A function is continuous on an interval if it is continuous at every point in the interval.

Concrete Examples:

Example 1: Consider the function f(x) = x². Is f(x) continuous at x = 2?

Setup: We need to check the three conditions for continuity.
Process:
1. f(2) = 2² = 4 (f(2) is defined).
2. lim (x→2) x² = 4 (the limit exists).
3. lim (x→2) x² = 4 = f(2) (the limit equals the function value).
Result: Since all three conditions are met, f(x) = x² is continuous at x = 2.
Why this matters: This shows a simple example of a continuous function.

Example 2: Consider the function h(x) = { x if x < 1, 2 if x ≥ 1 }. Is h(x) continuous at x = 1?

Setup: We need to check the three conditions for continuity.
Process:
1. h(1) = 2 (h(1) is defined).
2. lim (x→1-) h(x) = 1 (limit from the left is 1). lim (x→1+) h(x) = 2 (limit from the right is 2). Since the left and right limits are not equal, lim (x→1) h(x) does not exist.
3. Since the limit does not exist, the third condition is not met.
Result: h(x) is discontinuous at x = 1. This is a jump discontinuity.
Why this matters: This shows an example of a jump discontinuity where the left and right limits are different.

Analogies & Mental Models:

Think of it like a road: A continuous road is smooth and has no potholes or sudden jumps. A discontinuous road has potholes, sudden jumps, or is completely broken.
Where the analogy breaks down: This analogy is good for visualizing continuity, but it doesn't capture the formal mathematical definition.

Common Misconceptions:

❌ Students often think that if a function is defined at a point, it must be continuous at that point.
✓ A function can be defined at a point but still be discontinuous if the limit doesn't exist or doesn't equal the function value.
Why this confusion happens: It's easy to focus on the first condition (f(a) is defined) and forget about the other two.

Visual Description:

Imagine the graph of a function. A continuous function has no breaks, jumps, or holes in its graph. A discontinuous function has one or more of these features.

Practice Check:

Explain the three conditions for a function to be continuous at a point.

Connection to Other Sections: This section introduces the concept of continuity. The next section will explore the Intermediate Value Theorem, which is a powerful theorem that applies to continuous functions.

### 4.6 Intermediate Value Theorem (IVT)

*

Okay, here is a comprehensive lesson on Limits and Continuity in Calculus, designed for high school students (grades 9-12) with a focus on depth, clarity, and real-world applications.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're designing a roller coaster. You want it to be thrilling, but also safe. At the peak of a large hill, you need to ensure the track smoothly transitions from uphill to downhill. A sudden, jarring change in direction could be disastrous! To design this transition, you need to understand how the slope of the track approaches a specific point, and how the track continues without any breaks or jumps. This is where the concepts of limits and continuity come in.

Think about a dimmer switch on a light. As you slowly turn the knob, the light gradually brightens or dims. The brightness approaches a certain level as you approach a certain position on the switch. But what happens if the switch is broken, and there's a "dead spot" where the light suddenly jumps in brightness? That's a discontinuity. Limits and continuity help us understand these smooth transitions and identify potential problems.

### 1.2 Why This Matters

Limits and continuity are fundamental building blocks of calculus. Without them, we can't understand derivatives (rates of change) or integrals (areas under curves). These concepts are not just abstract mathematical ideas; they have immense real-world applications. Engineers use them to design bridges and airplanes, economists use them to model market behavior, computer scientists use them to create smooth animations and realistic simulations, and physicists use them to describe the motion of objects. Understanding limits and continuity opens doors to numerous STEM fields and provides a powerful tool for problem-solving. It builds directly upon your understanding of functions and graphs from algebra and pre-calculus. This knowledge will be essential for your future studies in calculus, differential equations, and beyond.

### 1.3 Learning Journey Preview

In this lesson, we will embark on a journey to explore the fascinating world of limits and continuity. We'll start by understanding what a limit is intuitively, then formalize the definition. We'll learn how to calculate limits using various techniques, including algebraic manipulation and graphical analysis. Next, we'll delve into the concept of continuity, exploring different types of discontinuities and how to identify them. We'll then connect limits and continuity, showing how continuity is defined in terms of limits. Finally, we'll examine real-world applications of these concepts and explore career paths where they are essential. Each concept builds upon the previous one, creating a cohesive understanding of these powerful mathematical tools.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

Explain the intuitive meaning of a limit and its connection to the behavior of a function near a point.
Calculate limits of functions using algebraic techniques, such as factoring, rationalizing, and simplifying.
Determine the existence of a limit graphically by analyzing the behavior of a function's graph as it approaches a specific point.
Define continuity of a function at a point and over an interval, and identify different types of discontinuities (removable, jump, infinite).
Apply the limit definition of continuity to determine if a function is continuous at a given point.
Analyze real-world scenarios involving rates of change and accumulation to identify where limits and continuity are applicable.
Evaluate the Intermediate Value Theorem and apply it to determine the existence of a root within a given interval.
Synthesize your understanding of limits and continuity to solve complex problems involving function behavior and real-world applications.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into limits and continuity, you should have a solid understanding of the following concepts:

Functions: Understanding what a function is (a relationship between inputs and outputs), function notation (f(x)), and different types of functions (linear, quadratic, polynomial, rational, trigonometric, exponential, logarithmic).
Graphs of Functions: Be able to graph functions and interpret their behavior from their graphs. Know how to identify key features like intercepts, asymptotes, and domain/range.
Algebraic Manipulation: Proficiency in simplifying expressions, factoring polynomials, solving equations, and working with inequalities.
Coordinate Geometry: Familiarity with the Cartesian coordinate system, distance formula, and slope of a line.
Trigonometry: Basic trigonometric functions (sine, cosine, tangent) and their properties.

If you need to review any of these concepts, refer to your algebra and pre-calculus textbooks or online resources like Khan Academy. A quick review of function notation and graphing techniques is especially helpful.

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## 4. MAIN CONTENT

### 4.1 What is a Limit? An Intuitive Introduction

Overview: The concept of a limit is fundamental to calculus. It describes the value that a function approaches as the input (x-value) approaches a certain value. It's not necessarily the value of the function at that point, but rather what the function is "heading towards."

The Core Concept: Imagine you're walking towards a building. As you get closer, your distance to the building approaches zero. The limit is similar: it's the value a function "gets closer and closer to" as the input gets closer and closer to a specific value. Crucially, the limit doesn't care what happens at the specific value itself. The function might be defined there, undefined there, or even defined at a completely different value. The limit is all about the function's behavior nearby.

We write the limit of f(x) as x approaches 'a' as: lim (x→a) f(x) = L. This means that as x gets arbitrarily close to 'a' (but not necessarily equal to 'a'), the value of f(x) gets arbitrarily close to L. "Arbitrarily close" is important. It means we can make f(x) as close to L as we want by choosing x sufficiently close to 'a'.

It's important to understand that the limit might not exist. If the function approaches different values from the left and the right, or if the function oscillates wildly near 'a', then the limit does not exist.

Concrete Examples:

Example 1: A Simple Polynomial
Setup: Consider the function f(x) = x + 2. We want to find the limit of f(x) as x approaches 3.
Process: As x gets closer to 3 (e.g., 2.9, 2.99, 2.999 or 3.1, 3.01, 3.001), f(x) gets closer to 3 + 2 = 5.
Result: lim (x→3) (x + 2) = 5
Why this matters: This demonstrates a straightforward limit where the function's value at the point is equal to the limit.

Example 2: A Rational Function with a Hole
Setup: Consider the function g(x) = (x^2 - 1) / (x - 1). Notice that g(1) is undefined (division by zero). However, we can simplify the function: g(x) = (x + 1)(x - 1) / (x - 1) = x + 1, for x ≠ 1.
Process: As x gets closer to 1 (but is not equal to 1), g(x) gets closer to 1 + 1 = 2.
Result: lim (x→1) (x^2 - 1) / (x - 1) = 2
Why this matters: This illustrates that the limit can exist even if the function is undefined at the point. The limit describes the function's behavior near the point, not at the point.

Analogies & Mental Models:

Think of it like... a GPS guiding you to a destination. The GPS tells you where to go next to reach your destination, even if you haven't arrived yet. The limit is like the GPS's prediction of where the function is "going" as x gets closer to a certain value.
[Explain how the analogy maps to the concept]: The GPS guides you towards the destination, even if there's a roadblock right at the final location. Similarly, the limit tells us where the function is heading, even if the function is undefined or has a different value at the specific input.
[Where the analogy breaks down (limitations)]: The GPS might be wrong due to traffic or construction. Similarly, the function might oscillate wildly and not actually "settle down" to a specific value, meaning the limit doesn't exist.

Common Misconceptions:

❌ Students often think that the limit is the same as the function's value at the point.
✓ Actually, the limit describes the function's behavior near the point, not necessarily at the point. The function might be undefined, or have a different value.
Why this confusion happens: Because in many simple cases (like continuous functions), the limit is equal to the function's value. It's easy to generalize this incorrectly.

Visual Description:

Imagine a graph of a function. The limit as x approaches 'a' is the y-value that the graph appears to be approaching as you trace the graph from both the left and the right sides towards the x-value 'a'. Even if there's a hole in the graph at x = a, the limit can still exist if the graph approaches a specific y-value from both sides.

Practice Check:

What is the limit of f(x) = 2x - 1 as x approaches 4?

Answer: lim (x→4) (2x - 1) = 2(4) - 1 = 7. As x gets closer to 4, 2x - 1 gets closer to 7.

Connection to Other Sections:

This section provides the intuitive foundation for understanding limits. The following sections will formalize this definition and introduce techniques for calculating limits. We will also see how this understanding is critical for defining continuity.

### 4.2 The Formal Definition of a Limit (ε-δ Definition)

Overview: While the intuitive understanding of a limit is helpful, we need a precise, mathematical definition to work with rigorously. This is provided by the epsilon-delta (ε-δ) definition of a limit.

The Core Concept: The ε-δ definition formalizes the idea of "arbitrarily close." It states that for any desired level of closeness (ε > 0) to the limit L, we can find a small enough interval around 'a' (δ > 0) such that all x-values within that interval (except possibly x = a) produce function values f(x) that are within ε of L.

Formally: lim (x→a) f(x) = L if and only if for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Let's break this down:

ε (epsilon): Represents the allowable error or tolerance. It's how close we want f(x) to be to L.
δ (delta): Represents the interval around 'a'. It's how close x needs to be to 'a' to ensure that f(x) is within ε of L.
|x - a| < δ: This means that the distance between x and 'a' is less than δ. In other words, x is within the interval (a - δ, a + δ), excluding a itself.
|f(x) - L| < ε: This means that the distance between f(x) and L is less than ε. In other words, f(x) is within the interval (L - ε, L + ε).
0 < |x - a|: Ensures that we are not considering x = a itself.

The definition essentially says: "No matter how small you make ε (the tolerance), I can always find a δ (interval around 'a') that guarantees f(x) is within that tolerance of L."

Concrete Examples:

Example 1: Proving a Simple Limit
Setup: Let's prove that lim (x→2) 3x - 1 = 5 using the ε-δ definition.
Process:
1. We want to show that for any ε > 0, there exists a δ > 0 such that if 0 < |x - 2| < δ, then |(3x - 1) - 5| < ε.
2. Simplify the inequality: |(3x - 1) - 5| = |3x - 6| = 3|x - 2|.
3. We want 3|x - 2| < ε. Divide both sides by 3: |x - 2| < ε/3.
4. Choose δ = ε/3.
5. Now, if 0 < |x - 2| < δ = ε/3, then 3|x - 2| < 3(ε/3) = ε. Therefore, |(3x - 1) - 5| < ε.
Result: We have shown that for any ε > 0, we can find a δ = ε/3 that satisfies the ε-δ definition. Therefore, lim (x→2) 3x - 1 = 5.
Why this matters: This demonstrates how to use the formal definition to rigorously prove a limit.

Example 2: A More Complex Function (Conceptual Only)
Setup: Proving limits for more complex functions like x^2 or sin(x) requires more sophisticated algebraic techniques to find the appropriate δ for a given ε.
Process: The general strategy is the same: manipulate the inequality |f(x) - L| < ε to isolate |x - a| and express it in terms of ε. The resulting expression will give you a suitable choice for δ.
Result: While the process can be complicated, the underlying principle remains the same: for any desired level of closeness, you can find a corresponding interval around 'a'.
Why this matters: Understanding the process, even if you don't perform the calculations, reinforces the meaning of the ε-δ definition.

Analogies & Mental Models:

Think of it like... a game of "how close can you get?" You want to get within a certain distance (ε) of a target. The other player gets to choose the target and the distance. Your goal is to always be able to find a way to get close enough (δ) to the target to win the game.
[Explain how the analogy maps to the concept]: The target is the limit L, and the distance is ε. Your move is to choose a range around 'a' (δ) such that your function value is always within ε of L. If you can always do this, the limit exists.
[Where the analogy breaks down (limitations)]: The mathematical definition is much more precise than a simple game. It requires a rigorous proof that the chosen δ works for every possible ε.

Common Misconceptions:

❌ Students often find the ε-δ definition confusing because of the abstract nature of ε and δ.
✓ Actually, ε and δ are just variables representing small distances. The definition simply formalizes the idea of "arbitrarily close."
Why this confusion happens: The notation and the quantifiers ("for every," "there exists") can be overwhelming at first.

Visual Description:

Imagine a graph of a function. Draw a horizontal band of width 2ε around the line y = L. The ε-δ definition says that you can always find a vertical band of width 2δ around the line x = a such that the portion of the graph within the vertical band also lies within the horizontal band. This holds true no matter how small you make ε.

Practice Check:

Explain in your own words what the ε-δ definition of a limit means.

Answer: The ε-δ definition says that we can make the output of a function as close as we want to a certain value (L) by making the input close enough to a specific value (a). We can always find an interval around 'a' such that all the function values within that interval are within our desired closeness to L.

Connection to Other Sections:

This section provides the formal definition of a limit. While it can be challenging to grasp at first, it is essential for a rigorous understanding of calculus. The next section will introduce techniques for calculating limits without directly using the ε-δ definition.

### 4.3 Techniques for Calculating Limits

Overview: While the ε-δ definition provides a rigorous foundation, it's not practical for calculating limits in most cases. Fortunately, we have several techniques that allow us to find limits more easily.

The Core Concept: These techniques rely on properties of limits and algebraic manipulation. Some common techniques include:

Direct Substitution: If f(x) is a continuous function at x = a, then lim (x→a) f(x) = f(a). This is the simplest case. Just plug in the value of 'a' into the function.
Factoring: If direct substitution results in an indeterminate form (0/0), try factoring the numerator and/or denominator to cancel out common factors.
Rationalizing: If the function involves radicals, try rationalizing the numerator or denominator to eliminate the indeterminate form.
Simplifying Complex Fractions: Simplify complex fractions before attempting to evaluate the limit.
L'Hôpital's Rule: If the limit is of the form 0/0 or ∞/∞, L'Hôpital's Rule states that lim (x→a) f(x) / g(x) = lim (x→a) f'(x) / g'(x), provided the limit on the right exists. (This requires knowledge of derivatives).
Squeeze Theorem (Sandwich Theorem): If g(x) ≤ f(x) ≤ h(x) for all x near 'a' (except possibly at 'a'), and lim (x→a) g(x) = lim (x→a) h(x) = L, then lim (x→a) f(x) = L.

Concrete Examples:

Example 1: Factoring
Setup: Find lim (x→2) (x^2 - 4) / (x - 2). Direct substitution gives 0/0.
Process: Factor the numerator: (x^2 - 4) = (x + 2)(x - 2). Then, (x^2 - 4) / (x - 2) = (x + 2)(x - 2) / (x - 2) = x + 2, for x ≠ 2.
Result: lim (x→2) (x^2 - 4) / (x - 2) = lim (x→2) (x + 2) = 2 + 2 = 4.
Why this matters: Factoring allows us to remove the indeterminate form and evaluate the limit.

Example 2: Rationalizing
Setup: Find lim (x→0) (√(x + 1) - 1) / x. Direct substitution gives 0/0.
Process: Multiply the numerator and denominator by the conjugate of the numerator: (√(x + 1) + 1). This gives: [(√(x + 1) - 1)(√(x + 1) + 1)] / [x(√(x + 1) + 1)] = (x + 1 - 1) / [x(√(x + 1) + 1)] = x / [x(√(x + 1) + 1)] = 1 / (√(x + 1) + 1), for x ≠ 0.
Result: lim (x→0) (√(x + 1) - 1) / x = lim (x→0) 1 / (√(x + 1) + 1) = 1 / (√(0 + 1) + 1) = 1/2.
Why this matters: Rationalizing allows us to eliminate the radical and evaluate the limit.

Example 3: L'Hôpital's Rule (Requires Derivatives - Preview for Later)
Setup: Find lim (x→0) sin(x)/x. Direct substitution gives 0/0.
Process: Applying L'Hôpital's Rule, we take the derivative of the numerator and denominator: lim (x→0) cos(x)/1
Result: lim (x→0) cos(x)/1 = cos(0)/1 = 1/1 = 1. Therefore, lim (x→0) sin(x)/x = 1.
Why this matters: L'Hôpital's Rule provides a powerful tool for evaluating limits of indeterminate forms when you know derivatives.

Analogies & Mental Models:

Think of it like... cleaning up a messy room. You need to identify the source of the mess (the indeterminate form) and then use appropriate tools (factoring, rationalizing) to tidy it up and reveal the underlying structure (the limit).
[Explain how the analogy maps to the concept]: The indeterminate form is the mess that prevents you from seeing the true value of the limit. The algebraic techniques are the cleaning tools that help you remove the mess.
[Where the analogy breaks down (limitations)]: Sometimes, cleaning up a room doesn't reveal anything useful. Similarly, sometimes algebraic manipulation doesn't lead to a solvable limit.

Common Misconceptions:

❌ Students often try to apply direct substitution even when it leads to an indeterminate form.
✓ Actually, direct substitution only works if the function is continuous at the point. If you get an indeterminate form, you need to use other techniques.
Why this confusion happens: Direct substitution is the easiest technique, so students often try it first without checking if it's valid.

Visual Description:

When factoring or rationalizing, you are essentially removing a "hole" or discontinuity from the graph of the function. The limit represents the y-value that the graph would have had at that point if the hole wasn't there.

Practice Check:

Find the limit of lim (x→-3) (x^2 + x - 6) / (x + 3).

Answer: First, try direct substitution, which gives 0/0. Factor the numerator: (x^2 + x - 6) = (x + 3)(x - 2). Then, (x^2 + x - 6) / (x + 3) = (x + 3)(x - 2) / (x + 3) = x - 2, for x ≠ -3. Therefore, lim (x→-3) (x^2 + x - 6) / (x + 3) = lim (x→-3) (x - 2) = -3 - 2 = -5.

Connection to Other Sections:

This section provides practical techniques for calculating limits. The next section will explore the concept of continuity, which is closely related to the existence and value of a limit.

### 4.4 One-Sided Limits

Overview: Sometimes, a function behaves differently as you approach a point from the left versus from the right. In these cases, we need to consider one-sided limits.

The Core Concept: A one-sided limit considers the behavior of a function as x approaches a value 'a' only from one direction: either from the left (x < a) or from the right (x > a).

Left-Hand Limit: The limit of f(x) as x approaches 'a' from the left is written as lim (x→a-) f(x) = L. This means that as x gets arbitrarily close to 'a' from values less than 'a', the value of f(x) gets arbitrarily close to L.
Right-Hand Limit: The limit of f(x) as x approaches 'a' from the right is written as lim (x→a+) f(x) = L. This means that as x gets arbitrarily close to 'a' from values greater than 'a', the value of f(x) gets arbitrarily close to L.

For a limit to exist (the two-sided limit we've discussed so far), both the left-hand limit and the right-hand limit must exist and be equal. If lim (x→a-) f(x) = L and lim (x→a+) f(x) = L, then lim (x→a) f(x) = L. Conversely, if the left-hand and right-hand limits are not equal, then the two-sided limit does not exist.

Concrete Examples:

Example 1: A Piecewise Function
Setup: Consider the piecewise function:
f(x) = { x + 1, if x < 2
{ 3x - 2, if x ≥ 2
Process:
Left-hand limit: lim (x→2-) f(x) = lim (x→2-) (x + 1) = 2 + 1 = 3.
Right-hand limit: lim (x→2+) f(x) = lim (x→2+) (3x - 2) = 3(2) - 2 = 4.
Result: Since the left-hand limit (3) is not equal to the right-hand limit (4), the limit lim (x→2) f(x) does not exist.
Why this matters: This demonstrates that one-sided limits are crucial for determining the existence of a two-sided limit, especially for piecewise functions.

Example 2: A Function with a Vertical Asymptote
Setup: Consider the function f(x) = 1/x. We want to find the one-sided limits as x approaches 0.
Process:
Left-hand limit: lim (x→0-) f(x) = lim (x→0-) (1/x) = -∞. As x approaches 0 from the left (negative values), 1/x becomes increasingly negative.
Right-hand limit: lim (x→0+) f(x) = lim (x→0+) (1/x) = ∞. As x approaches 0 from the right (positive values), 1/x becomes increasingly positive.
Result: Since both one-sided limits approach infinity (albeit in different directions), we still say that the limit lim (x→0) (1/x) does not exist.
Why this matters: This illustrates how one-sided limits can help understand the behavior of functions near vertical asymptotes.

Analogies & Mental Models:

Think of it like... approaching a fork in the road. You can either go left or right. The one-sided limits are like the destinations you reach depending on which direction you choose. If the destinations are different, there's no single destination for you to approach.
[Explain how the analogy maps to the concept]: The fork in the road represents the point 'a'. The left and right paths represent the left-hand and right-hand limits. If the paths lead to different places, the overall destination is undefined.
[Where the analogy breaks down (limitations)]: The analogy doesn't capture the idea of "arbitrarily close." In mathematics, we need to consider the behavior of the function as x gets infinitely close to 'a' from both sides.

Common Misconceptions:

❌ Students often think that if a function is undefined at a point, the limit automatically doesn't exist.
✓ Actually, the limit might still exist if the left-hand and right-hand limits are equal, even if the function is undefined at the point.
Why this confusion happens: The limit describes the function's behavior near the point, not necessarily at the point.

Visual Description:

When looking at the graph of a function, examine what y-value the function approaches as you trace the graph from the left towards x = a, and then separately from the right towards x = a. If the y-values are different, the overall limit doesn't exist.

Practice Check:

Consider the following function:

f(x) = { x^2, if x ≤ 1
{ 2x, if x > 1

Does the limit of f(x) as x approaches 1 exist?

Answer: Left-hand limit: lim (x→1-) f(x) = lim (x→1-) x^2 = 1^2 = 1. Right-hand limit: lim (x→1+) f(x) = lim (x→1+) 2x = 2(1) = 2. Since the left-hand limit (1) is not equal to the right-hand limit (2), the limit lim (x→1) f(x) does not exist.

Connection to Other Sections:

This section builds on the previous discussion of limits by introducing the concept of one-sided limits. This is crucial for understanding continuity, which we will explore in the next section.

### 4.5 Continuity

Overview: Continuity is a fundamental property of functions that describes whether a function's graph can be drawn without lifting your pen. It's closely related to the concept of limits.

The Core Concept: A function f(x) is said to be continuous at a point x = a if the following three conditions are met:

1. f(a) is defined: The function must be defined at the point x = a. There is a value for f(a).
2. lim (x→a) f(x) exists: The limit of f(x) as x approaches 'a' must exist. This means that both the left-hand limit and the right-hand limit must exist and be equal.
3. lim (x→a) f(x) = f(a): The limit of f(x) as x approaches 'a' must be equal to the function's value at 'a'.

If any of these three conditions are not met, the function is said to be discontinuous at x = a.

A function is continuous on an interval if it is continuous at every point in that interval.

Concrete Examples:

Example 1: A Continuous Polynomial Function
Setup: Consider the function f(x) = x^2 + 1. Let's check if it's continuous at x = 2.
Process:
1. f(2) = 2^2 + 1 = 5 (defined).
2. lim (x→2) f(x) = lim (x→2) (x^2 + 1) = 2^2 + 1 = 5 (exists).
3. lim (x→2) f(x) = f(2) = 5.
Result: All three conditions are met, so f(x) is continuous at x = 2. In fact, polynomial functions are continuous everywhere.
Why this matters: This illustrates a simple example of a continuous function.

Example 2: A Discontinuous Piecewise Function
Setup: Consider the piecewise function:
f(x) = { x + 1, if x < 2
{ 3x - 2, if x ≥ 2
Process: As we saw in the one-sided limits example, lim (x→2-) f(x) = 3 and lim (x→2+) f(x) = 4. Therefore, lim (x→2) f(x) does not exist.
Result: Since the limit does not exist, f(x) is discontinuous at x = 2.
Why this matters: This illustrates a function that fails the second condition for continuity.

Example 3: A Function with a Removable Discontinuity
Setup: Consider the function g(x) = (x^2 - 1) / (x - 1), for x ≠ 1, and g(1) = 3.
Process:
1. g(1) = 3 (defined).
2. lim (x→1) g(x) = lim (x→1) (x + 1) = 2 (exists).
3. lim (x→1) g(x) = 2 ≠ g(1) = 3.
Result: The third condition is not met, so g(x) is discontinuous at x = 1. This is called a removable discontinuity because if we redefined g(1) to be 2, the function would become continuous.
Why this matters: This illustrates a function that fails the third condition for continuity.

Analogies & Mental Models:

Think of it like... a continuous road vs. a road with a gap. A continuous road allows you to travel smoothly from one point to another without any interruptions. A road with a gap requires you to jump over the gap, making the journey discontinuous.
[Explain how the analogy maps to the concept]: The continuous road represents a continuous function. The gap in the road represents a discontinuity.
[Where the analogy breaks down (limitations)]: The analogy doesn't capture the different types of discontinuities (removable, jump, infinite).

Common Misconceptions:

❌ Students often think that if a function is defined at a point, it is automatically continuous at that point.
✓ Actually, the limit must also exist and be equal to the function's value at the point.
Why this confusion happens: Because it’s easy to forget to check all three conditions for continuity.

Visual Description:

A continuous function has a graph that you can draw without lifting your pen. A discontinuous function has a break, jump, or hole in its graph.

Practice Check:

Is the function f(x) = 1/x continuous at x = 0? Why or why not?

Answer: No, f(x) is not continuous at x = 0 because f(0) is undefined. The first condition for continuity is not met.

Connection to Other Sections:

This section defines continuity in terms of limits. The next section will explore different types of discontinuities and how to identify them.

### 4.6 Types of Discontinuities

Overview: When a function is discontinuous at a point, it can be discontinuous in several ways. Understanding the different types of discontinuities is important for analyzing function behavior.

The Core Concept: There are three main types of discontinuities:

1. Removable Discontinuity: A removable discontinuity occurs when the limit of the function exists at the point, but either the function is undefined at the point or the limit is not equal to the function's value. This type of discontinuity can be "removed" by redefining the function at that point to be equal to the limit.
2. Jump Discontinuity: A jump discontinuity occurs when the left-hand limit and the right-hand limit both exist, but they are not equal. This creates a "jump" in the graph of the function.
3. Infinite Dis

Okay, here's a comprehensive lesson on Limits and Continuity in Calculus, designed to be engaging, thorough, and accessible for high school students.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're designing a roller coaster. You want it to be thrilling, but also safe. You need to know how close the coaster can get to a loop-de-loop without flying off the tracks. Or, think about zooming in on a digital photo. At first, it looks smooth, but as you zoom closer and closer, you start to see the individual pixels. What happens infinitely close? These questions get at the heart of what limits are all about: understanding what happens to a function as we get arbitrarily close to a particular point. Limits are not just an abstract mathematical concept; they are the foundation upon which calculus is built, and they help us understand change, approximation, and behavior at extreme scales. We're going to explore how this powerful tool works.

### 1.2 Why This Matters

Limits form the bedrock of calculus, which is essential for understanding physics, engineering, economics, computer science, and many other fields. Without limits, we couldn't define derivatives (rates of change) or integrals (areas under curves) – the core operations of calculus. Think about designing bridges (ensuring structural integrity under extreme loads), predicting weather patterns (modeling complex atmospheric changes), or creating realistic animations in video games (approximating smooth curves with polygons). All of these rely on calculus, and therefore, on limits. This lesson builds on your prior knowledge of functions and algebra, providing the foundation for understanding derivatives, integrals, and differential equations in your future math courses. It’s the first step towards unlocking the power to model and solve complex problems in the real world.

### 1.3 Learning Journey Preview

In this lesson, we will first define what a limit is and explore different ways to evaluate them. We will learn about graphical, numerical, and algebraic approaches. We'll then discuss continuity, which is closely related to limits. We will learn about the different types of discontinuities and how to identify them. We will see how limits and continuity are used in real-world applications, from physics to computer graphics. Finally, we'll look at some more advanced topics, such as infinite limits and limits at infinity. Each concept will be built upon previous ones, so you can gradually develop a solid understanding of limits and continuity.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

Define the concept of a limit of a function at a point, both informally and formally (using the epsilon-delta definition).
Evaluate limits graphically, numerically (using tables of values), and algebraically (using limit laws and simplification techniques).
Identify and classify different types of discontinuities (removable, jump, infinite) in a function.
Apply the concept of continuity to determine if a function is continuous at a given point and over an interval.
Explain the Intermediate Value Theorem and use it to prove the existence of a root of a function within a given interval.
Evaluate limits involving infinity (both infinite limits and limits at infinity) and interpret their meaning in terms of asymptotes.
Solve real-world problems involving limits and continuity, such as finding the instantaneous velocity of an object or modeling continuous growth.
Synthesize the relationship between limits and continuity and explain how they form the foundation of calculus.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into limits and continuity, you should be comfortable with the following:

Functions: Understanding what a function is, how to represent it (graphically, algebraically, numerically), and concepts like domain and range.
Algebraic manipulation: Simplifying expressions, solving equations, factoring, and working with inequalities.
Graphing: Being able to graph basic functions (linear, quadratic, polynomial, rational, trigonometric) and understand their properties.
Trigonometry: Knowing the definitions of trigonometric functions (sine, cosine, tangent) and their basic identities.

Quick Review:

Functions: A function is a rule that assigns to each input value (x) exactly one output value (f(x)).
Domain: The set of all possible input values (x) for which a function is defined.
Range: The set of all possible output values (f(x)) that a function can take.
Factoring: Breaking down an expression into a product of simpler expressions (e.g., x² - 4 = (x-2)(x+2)).
Inequalities: Mathematical statements comparing two expressions using symbols like <, >, ≤, ≥.

If you need to refresh your understanding of any of these topics, review your algebra and precalculus textbooks or online resources like Khan Academy. A solid foundation in these areas will make learning limits and continuity much easier.

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## 4. MAIN CONTENT

### 4.1 Introduction to Limits: The Idea of Approaching a Value

Overview: The concept of a limit is about understanding the behavior of a function as its input approaches a specific value. It's not necessarily about what the function is at that value, but rather what it approaches.

The Core Concept:

Imagine you're walking towards a destination. You get closer and closer, but you don't necessarily have to reach the destination to understand where you're heading. That's the idea behind a limit. Formally, the limit of a function f(x) as x approaches a value 'a' is denoted as:

lim_x→a f(x) = L

This means that as x gets arbitrarily close to 'a' (but not necessarily equal to 'a'), the value of f(x) gets arbitrarily close to L. 'L' is the limit.

It's crucial to understand that the limit doesn't care about what happens at x = a. The function might be defined at x = a, it might not be, or it might have a completely different value than L. The limit only describes the behavior of f(x) near x = a.

We need to consider approaching 'a' from both sides – from values less than 'a' (the left-hand limit) and from values greater than 'a' (the right-hand limit). For the limit to exist, both the left-hand limit and the right-hand limit must exist and be equal.

lim_x→a^- f(x) = L (Left-hand limit)
lim_x→a⁺ f(x) = L (Right-hand limit)

If these two limits are equal, then the overall limit exists and is equal to L. If they are different, the limit does not exist (DNE).

Concrete Examples:

Example 1: Consider the function f(x) = (x² - 1) / (x - 1).

Setup: We want to find the limit of f(x) as x approaches 1. Notice that f(1) is undefined because it would result in division by zero.
Process: We can simplify the function by factoring the numerator: f(x) = (x - 1)(x + 1) / (x - 1). For x ≠ 1, we can cancel the (x - 1) terms, so f(x) = x + 1. Now, as x approaches 1, x + 1 approaches 1 + 1 = 2.
Result: lim_x→1 (x² - 1) / (x - 1) = 2. Even though the function is undefined at x = 1, the limit as x approaches 1 is 2.
Why this matters: This example shows that a limit can exist even if the function is not defined at the point it's approaching.

Example 2: Consider the function g(x) =

x if x < 2
5 if x ≥ 2

Setup: We want to find the limit of g(x) as x approaches 2.
Process: We need to consider the left-hand and right-hand limits.
Left-hand limit: lim_x→2^- g(x) = lim_x→2^- x = 2. As x approaches 2 from the left, g(x) approaches 2.
Right-hand limit: lim_x→2⁺ g(x) = lim_x→2⁺ 5 = 5. As x approaches 2 from the right, g(x) approaches 5.
Result: Since the left-hand limit (2) and the right-hand limit (5) are not equal, the limit of g(x) as x approaches 2 does not exist.
Why this matters: This example demonstrates that for a limit to exist, the function must approach the same value from both sides.

Analogies & Mental Models:

Think of it like... aiming a laser pointer at a target. The limit is where you're aiming, even if you don't actually hit the target. The function is the path of the laser, and the target is the point 'a'. The limit exists if the laser gets arbitrarily close to the target.
Limitations: The analogy breaks down if the laser pointer is shaky and jumps around wildly. In that case, the limit might not exist.

Common Misconceptions:

❌ Students often think... the limit is the value of the function at x = a.
✓ Actually... the limit is the value the function approaches as x gets arbitrarily close to a, regardless of the function's value at a.
Why this confusion happens: Because in many simple cases, the limit is equal to the function's value at that point. But this is not always true.

Visual Description:

Imagine a graph of a function. To find the limit as x approaches 'a', focus on the region near x = a. Trace the graph from the left and from the right, getting closer and closer to x = a. If the y-values of the graph converge to a single value 'L' from both sides, then the limit exists and is equal to L. If the y-values approach different values from the left and right, the limit does not exist.

Practice Check:

Find the limit of f(x) = 3x + 2 as x approaches 1.

Answer with explanation: lim_x→1 (3x + 2) = 3(1) + 2 = 5. In this case, the function is continuous, so the limit is simply the value of the function at x = 1.

Connection to Other Sections: This section lays the foundation for all subsequent sections. Understanding the basic concept of a limit is crucial for understanding how to evaluate limits, define continuity, and apply these concepts to real-world problems.

### 4.2 Methods for Evaluating Limits: Graphical, Numerical, and Algebraic

Overview: There are several ways to determine the value of a limit: graphically, numerically (using tables), and algebraically (using limit laws and simplification techniques).

The Core Concept:

We can use different methods to find the limit of a function f(x) as x approaches a value 'a':

1. Graphical Method: Examine the graph of the function near x = a. Observe the behavior of the y-values as x approaches 'a' from both the left and right. If the graph approaches the same y-value 'L' from both sides, then the limit is L.

2. Numerical Method (Tables of Values): Create a table of x-values that are close to 'a' (both slightly less than and slightly greater than 'a') and calculate the corresponding f(x) values. Observe the trend of the f(x) values as x gets closer to 'a'. If the f(x) values approach a specific value 'L', then the limit is L.

3. Algebraic Method (Limit Laws and Simplification): Use algebraic techniques to simplify the function and then apply limit laws to evaluate the limit. Some common techniques include factoring, rationalizing the numerator or denominator, and using trigonometric identities. Limit laws allow you to break down complex limits into simpler ones.

Concrete Examples:

Example 1: Graphical Method

Setup: Consider the graph of a function f(x). We want to find the limit of f(x) as x approaches 2.
Process: By looking at the graph near x = 2, we can see that as x approaches 2 from the left, the y-values approach 3. Similarly, as x approaches 2 from the right, the y-values also approach 3.
Result: Therefore, lim_x→2 f(x) = 3.
Why this matters: This demonstrates how to visually estimate a limit using a graph.

Example 2: Numerical Method

Setup: Consider the function f(x) = sin(x)/x. We want to find the limit of f(x) as x approaches 0.
Process: Create a table of values:

| x | f(x) = sin(x)/x |
| --------- | --------------- |
| -0.1 | 0.99833 |
| -0.01 | 0.99998 |
| -0.001 | 0.9999998 |
| 0 | Undefined |
| 0.001 | 0.9999998 |
| 0.01 | 0.99998 |
| 0.1 | 0.99833 |

Result: As x gets closer to 0 from both sides, f(x) approaches 1. Therefore, lim_x→0 sin(x)/x = 1.
Why this matters: This demonstrates how to approximate a limit using a table of values.

Example 3: Algebraic Method

Setup: Consider the function f(x) = (x² + 3x + 2) / (x + 2). We want to find the limit of f(x) as x approaches -2.
Process: Factor the numerator: f(x) = (x + 1)(x + 2) / (x + 2). For x ≠ -2, we can cancel the (x + 2) terms, so f(x) = x + 1. Now, we can apply the limit law that the limit of a sum is the sum of the limits: lim_x→-2 (x + 1) = lim_x→-2 x + lim_x→-2 1 = -2 + 1 = -1.
Result: Therefore, lim_x→-2 (x² + 3x + 2) / (x + 2) = -1.
Why this matters: This demonstrates how to use algebraic simplification and limit laws to find a limit.

Analogies & Mental Models:

Think of it like... using different tools to measure the length of a table. You can use a ruler (algebraic), a measuring tape (numerical), or simply visually estimate it (graphical). Each tool has its advantages and disadvantages.
Limitations: The graphical method is limited by the accuracy of the graph. The numerical method is limited by the number of data points and the precision of the calculations. The algebraic method requires knowledge of limit laws and simplification techniques.

Common Misconceptions:

❌ Students often think... that the numerical method always gives the exact value of the limit.
✓ Actually... the numerical method provides an approximation of the limit. The closer the x-values are to 'a', the better the approximation.
Why this confusion happens: Because calculators and computers can give very precise approximations, it's easy to mistake them for exact values.

Visual Description:

Graphical: A graph where you trace the function from both sides towards a point.
Numerical: A table of numbers getting closer and closer to 'a' and observing the output values.
Algebraic: A series of algebraic manipulations that simplify the function.

Practice Check:

Evaluate the limit of f(x) = (x² - 4) / (x - 2) as x approaches 2 using the algebraic method.

Answer with explanation: Factor the numerator: f(x) = (x - 2)(x + 2) / (x - 2). For x ≠ 2, we can cancel the (x - 2) terms, so f(x) = x + 2. Therefore, lim_x→2 (x² - 4) / (x - 2) = lim_x→2 (x + 2) = 2 + 2 = 4.

Connection to Other Sections: This section provides the tools necessary to evaluate limits, which is essential for understanding continuity and applying limits to real-world problems. It builds on the previous section by providing concrete methods for finding limits.

### 4.3 Limit Laws: Rules for Simplifying Limits

Overview: Limit laws are a set of rules that allow us to break down complex limits into simpler ones, making them easier to evaluate.

The Core Concept:

Limit laws provide a systematic way to evaluate limits of combinations of functions. Here are some of the most important limit laws:

1. Limit of a Constant: lim_x→a c = c (The limit of a constant is the constant itself)
2. Limit of x: lim_x→a x = a (The limit of x as x approaches 'a' is 'a')
3. Limit of a Sum/Difference: lim_x→a [f(x) ± g(x)] = lim_x→a f(x) ± lim_x→a g(x) (The limit of a sum or difference is the sum or difference of the limits)
4. Limit of a Constant Multiple: lim_x→a [c f(x)] = c lim_x→a f(x) (The limit of a constant times a function is the constant times the limit of the function)
5. Limit of a Product: lim_x→a [f(x) g(x)] = lim_x→a f(x) lim_x→a g(x) (The limit of a product is the product of the limits)
6. Limit of a Quotient: lim_x→a [f(x) / g(x)] = [lim_x→a f(x)] / [lim_x→a g(x)] (provided lim_x→a g(x) ≠ 0) (The limit of a quotient is the quotient of the limits, as long as the limit of the denominator is not zero)
7. Limit of a Power: lim_x→a [f(x)]ⁿ = [lim_x→a f(x)]ⁿ (The limit of a function raised to a power is the limit of the function raised to that power)
8. Limit of a Root: lim_x→a ⁿ√[f(x)] = ⁿ√[lim_x→a f(x)] (provided the nth root of lim_x→a f(x) exists) (The limit of a root is the root of the limit, as long as the root exists)

Concrete Examples:

Example 1: Find the limit of f(x) = 3x² + 2x - 1 as x approaches 2.

Setup: We want to evaluate lim_x→2 (3x² + 2x - 1).
Process: Using the limit laws, we can break this down:
lim_x→2 (3x² + 2x - 1) = lim_x→2 (3x²) + lim_x→2 (2x) - lim_x→2 (1) (Limit of a sum/difference)
= 3 lim_x→2 (x²) + 2 lim_x→2 (x) - 1 (Limit of a constant multiple and limit of a constant)
= 3 (lim_x→2 x)² + 2 (2) - 1 (Limit of a power and limit of x)
= 3 (2)² + 4 - 1 = 3 4 + 4 - 1 = 12 + 4 - 1 = 15
Result: Therefore, lim_x→2 (3x² + 2x - 1) = 15.
Why this matters: This demonstrates how to use limit laws to evaluate the limit of a polynomial function.

Example 2: Find the limit of f(x) = √(x + 4) / (x - 2) as x approaches 5.

Setup: We want to evaluate lim_x→5 √(x + 4) / (x - 2).
Process: Using the limit laws, we can break this down:
lim_x→5 √(x + 4) / (x - 2) = [lim_x→5 √(x + 4)] / [lim_x→5 (x - 2)] (Limit of a quotient)
= √[lim_x→5 (x + 4)] / [lim_x→5 x - lim_x→5 2] (Limit of a root and limit of a sum/difference)
= √(5 + 4) / (5 - 2) = √9 / 3 = 3 / 3 = 1
Result: Therefore, lim_x→5 √(x + 4) / (x - 2) = 1.
Why this matters: This demonstrates how to use limit laws to evaluate the limit of a more complex function.

Analogies & Mental Models:

Think of it like... a recipe. Limit laws are like the individual steps in the recipe that allow you to combine ingredients (functions) in a specific order to create a final dish (the limit).
Limitations: Limit laws only work if the individual limits exist. If one of the limits does not exist, you cannot apply the limit laws. Also, remember the quotient rule only works if the limit of the denominator is not zero.

Common Misconceptions:

❌ Students often think... they can apply the limit laws even if the individual limits do not exist.
✓ Actually... the limit laws only apply if the individual limits exist.
Why this confusion happens: Because it's easy to blindly apply the limit laws without checking if the conditions are met.

Visual Description:

A diagram showing how a complex limit is broken down into simpler limits using the limit laws. Each limit law is represented by an arrow showing how the limit is transformed.

Practice Check:

Evaluate the limit of f(x) = (x³ - 8) / (x - 2) as x approaches 2 using the limit laws (after factoring).

Answer with explanation: First, factor the numerator: x³ - 8 = (x - 2)(x² + 2x + 4). Then, f(x) = (x - 2)(x² + 2x + 4) / (x - 2). For x ≠ 2, f(x) = x² + 2x + 4. Now, use the limit laws: lim_x→2 (x² + 2x + 4) = lim_x→2 x² + lim_x→2 2x + lim_x→2 4 = 2² + 2(2) + 4 = 4 + 4 + 4 = 12.

Connection to Other Sections: This section provides the algebraic tools needed to evaluate limits more efficiently. It builds on the previous section by providing specific rules for simplifying limits.

### 4.4 Continuity: What it Means for a Function to be "Unbroken"

Overview: Continuity is a fundamental concept in calculus that describes functions that are "unbroken" or "smooth."

The Core Concept:

A function f(x) is continuous at a point x = a if the following three conditions are met:

1. f(a) is defined: The function must be defined at x = a. In other words, 'a' must be in the domain of f.
2. lim_x→a f(x) exists: The limit of f(x) as x approaches 'a' must exist. This means that the left-hand limit and the right-hand limit must be equal.
3. lim_x→a f(x) = f(a): The limit of f(x) as x approaches 'a' must be equal to the value of the function at x = a.

If any of these three conditions are not met, then the function is discontinuous at x = a.

A function is continuous on an interval if it is continuous at every point in the interval.

Concrete Examples:

Example 1: Continuous Function

Setup: Consider the function f(x) = x². We want to determine if f(x) is continuous at x = 2.
Process:
1. f(2) = 2² = 4 (f(2) is defined)
2. lim_x→2 f(x) = lim_x→2 x² = 4 (The limit exists)
3. lim_x→2 f(x) = f(2) = 4 (The limit is equal to the function value)
Result: Since all three conditions are met, f(x) = x² is continuous at x = 2.
Why this matters: This demonstrates a simple example of a continuous function.

Example 2: Discontinuous Function

Setup: Consider the function g(x) =

x if x < 1
3 if x = 1
2x if x > 1

We want to determine if g(x) is continuous at x = 1.
Process:
1. g(1) = 3 (g(1) is defined)
2. lim_x→1^- g(x) = lim_x→1^- x = 1 (Left-hand limit)
3. lim_x→1⁺ g(x) = lim_x→1⁺ 2x = 2 (Right-hand limit)
4. Since the left-hand limit (1) and the right-hand limit (2) are not equal, lim_x→1 g(x) does not exist.
Result: Since the limit does not exist, g(x) is discontinuous at x = 1.
Why this matters: This demonstrates an example of a function that is discontinuous because the limit does not exist.

Analogies & Mental Models:

Think of it like... drawing a graph without lifting your pencil. If you can draw the graph of a function without lifting your pencil, then the function is continuous.
Limitations: This analogy breaks down for functions with infinitely many discontinuities.

Common Misconceptions:

❌ Students often think... that if a function is defined at a point, then it is continuous at that point.
✓ Actually... a function must satisfy all three conditions (defined, limit exists, limit equals function value) to be continuous.
Why this confusion happens: Because the first condition (f(a) is defined) is often mistakenly considered sufficient for continuity.

Visual Description:

Continuous: A smooth, unbroken curve.
Discontinuous: A graph with a break, jump, or hole.

Practice Check:

Determine if the function f(x) = 1/x is continuous at x = 0.

Answer with explanation: f(0) is undefined (division by zero). Therefore, the function is discontinuous at x = 0.

Connection to Other Sections: This section builds on the previous sections by using the concept of limits to define continuity. Understanding continuity is essential for many theorems in calculus, such as the Intermediate Value Theorem.

### 4.5 Types of Discontinuities: Removable, Jump, and Infinite

Overview: When a function is discontinuous, it can be discontinuous in different ways. Understanding these types of discontinuities helps us analyze the function's behavior.

The Core Concept:

There are three main types of discontinuities:

1. Removable Discontinuity: A removable discontinuity occurs when the limit of the function exists at x = a, but either f(a) is undefined or f(a) is not equal to the limit. This type of discontinuity can be "removed" by redefining the function at x = a to be equal to the limit.

2. Jump Discontinuity: A jump discontinuity occurs when the left-hand limit and the right-hand limit exist at x = a, but they are not equal. The function "jumps" from one value to another at x = a.

3. Infinite Discontinuity: An infinite discontinuity occurs when the function approaches infinity (or negative infinity) as x approaches 'a' from either the left or the right. This often occurs when there is a vertical asymptote at x = a.

Concrete Examples:

Example 1: Removable Discontinuity

Setup: Consider the function f(x) = (x² - 4) / (x - 2). This function is undefined at x = 2.
Process: We know that lim_x→2 (x² - 4) / (x - 2) = 4 (from a previous example). We can redefine the function as:

f(x) = (x² - 4) / (x - 2) if x ≠ 2
4 if x = 2

Result: By redefining the function, we have "removed" the discontinuity.
Why this matters: This demonstrates how a removable discontinuity can be fixed by redefining the function at the point of discontinuity.

Example 2: Jump Discontinuity

Setup: Consider the function g(x) =

1 if x < 0
0 if x = 0
-1 if x > 0

Process:
lim_x→0^- g(x) = 1
lim_x→0⁺ g(x) = -1
Since the left-hand limit and the right-hand limit are not equal, there is a jump discontinuity at x = 0.
Result: The function jumps from 1 to -1 at x = 0.
Why this matters: This demonstrates a clear example of a jump discontinuity.

Example 3: Infinite Discontinuity

Setup: Consider the function h(x) = 1/x².
Process: As x approaches 0 from either the left or the right, h(x) approaches infinity.
Result: There is an infinite discontinuity at x = 0. The graph has a vertical asymptote at x=0.
Why this matters: This demonstrates how a function with a vertical asymptote has an infinite discontinuity.

Analogies & Mental Models:

Think of it like... different types of cracks in a road. A removable discontinuity is like a small pothole that can be easily filled. A jump discontinuity is like a sudden drop-off in the road. An infinite discontinuity is like a sinkhole that swallows the road completely.
Limitations: This analogy is limited because mathematical discontinuities are precise and well-defined, while real-world cracks are more complex.

Common Misconceptions:

❌ Students often think... that all discontinuities are the same.
✓ Actually... there are different types of discontinuities, each with its own characteristics and behavior.
Why this confusion happens: Because the term "discontinuity" is a general term that encompasses different specific types.

Visual Description:

Removable: A graph with a hole that could be filled in.
Jump: A graph where the function jumps from one y-value to another.
Infinite: A graph with a vertical asymptote.

Practice Check:

Identify the type of discontinuity in the function f(x) = (x + 3) / (x² - 9) at x = -3.

Answer with explanation: First, factor the denominator: f(x) = (x + 3) / [(x + 3)(x - 3)]. For x ≠ -3, f(x) = 1 / (x - 3). The limit as x approaches -3 is 1/(-3-3) = -1/6. Since the limit exists, but the function is undefined at x = -3, this is a removable discontinuity.

Connection to Other Sections: This section builds on the previous section by classifying different types of discontinuities. Understanding the type of discontinuity helps us analyze the behavior of the function and determine how to deal with it.

### 4.6 The Intermediate Value Theorem (IVT): Guaranteeing Values

Overview: The Intermediate Value Theorem is a powerful theorem that guarantees the existence of a specific value within a continuous function.

The Core Concept:

The Intermediate Value Theorem (IVT) states that if f(x) is a continuous function on a closed interval [a, b], and k is any number between f(a) and f(b), then there exists at least one number c in the interval (a, b) such that f(c) = k.

In simpler terms, if a continuous function takes on two values, f(a) and f(b), then it must take on every value in between those two values at some point in the interval [a, b].

Concrete Examples:

Example 1:

Setup: Let f(x) = x² + x - 1. Show that there is a root (a value of x where f(x) = 0) between x = 0 and x = 1.

Okay, here's a comprehensive lesson on Limits and Continuity, designed for high school students with an emphasis on deeper understanding and applications. This is a substantial piece of content, so be prepared for a long read.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're designing a roller coaster. The goal is to create a thrilling ride, but you absolutely cannot have any sudden, jarring changes in direction or speed. A smooth, continuous track is essential. Or, consider a self-driving car navigating a busy street. It needs to predict the future position of other vehicles and pedestrians to avoid collisions. These predictions rely on understanding how things approach a certain state – their limit – and whether that approach is predictable and continuous. Finally, think about the stock market. While its behavior seems chaotic, algorithms attempt to predict future prices based on past trends. These algorithms are fundamentally built on the concepts of limits and continuity.

These seemingly disparate scenarios – roller coasters, self-driving cars, and the stock market – share a common mathematical foundation: limits and continuity. These aren't just abstract concepts; they are the bedrock of calculus and, by extension, many technologies and fields we rely on every day. We will discover that limits allow us to examine the behavior of functions near a specific point, even if the function isn't defined at that point. Continuity takes this further, ensuring that functions behave predictably and smoothly.

### 1.2 Why This Matters

Limits and continuity are not just theoretical exercises; they are essential for understanding real-world phenomena. They form the foundation upon which calculus is built. Without a firm grasp of these concepts, you'll struggle with derivatives, integrals, and the many applications of calculus. In engineering, understanding limits is crucial for designing stable structures and efficient systems. In computer science, limits play a role in analyzing the efficiency of algorithms. In economics, they are used to model growth and change. Furthermore, a strong understanding of these concepts opens doors to advanced topics in mathematics, such as real analysis and topology.

This lesson builds upon your existing knowledge of functions, graphs, and basic algebra. You'll be applying these skills to analyze the behavior of functions in a more sophisticated way. Mastering limits and continuity is a crucial stepping stone to more advanced calculus topics, such as differentiation and integration. These, in turn, will allow you to solve a wide range of problems in physics, engineering, economics, and other fields.

### 1.3 Learning Journey Preview

In this lesson, we'll begin by defining what a limit is and how to calculate it. We'll explore different techniques for finding limits, including direct substitution, factoring, rationalizing, and using limit laws. We'll then delve into the concept of continuity, examining its definition and different types of discontinuities. We'll learn how to determine if a function is continuous at a point and on an interval. Finally, we'll explore real-world applications of limits and continuity and discuss their importance in various fields. Each concept will be built upon the previous one, creating a solid foundation for your understanding of calculus. We will also touch on the history and the mathematicians that have contributed to the understanding of these concepts.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

Explain the formal definition of a limit and its intuitive meaning.
Calculate limits of functions using direct substitution, factoring, rationalizing, and limit laws.
Analyze one-sided limits and their relationship to the existence of a two-sided limit.
Identify and classify different types of discontinuities (removable, jump, infinite).
Determine if a function is continuous at a point and on an interval using the definition of continuity.
Apply the Intermediate Value Theorem to determine the existence of a root within a given interval.
Evaluate limits involving infinity and identify horizontal asymptotes.
Synthesize your understanding of limits and continuity to solve real-world problems in various fields such as physics and engineering.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into limits and continuity, you should have a solid understanding of the following:

Functions: Definition of a function, domain, range, independent and dependent variables, function notation (e.g., f(x)).
Graphs of Functions: Understanding how to graph various types of functions, including linear, quadratic, polynomial, rational, trigonometric, exponential, and logarithmic functions.
Algebraic Manipulation: Proficiency in algebraic techniques such as factoring, simplifying expressions, solving equations, and working with inequalities.
Trigonometry: Basic trigonometric functions (sine, cosine, tangent) and their properties.
Interval Notation: Understanding how to represent sets of real numbers using interval notation (e.g., (a, b), [a, b], (a, ∞)).

Quick Review:

Function: A relation between a set of inputs (domain) and a set of permissible outputs (range) with the property that each input is related to exactly one output.
Graph: A visual representation of a function, showing the relationship between the input (x-axis) and the output (y-axis).
Factoring: The process of breaking down an expression into a product of simpler expressions. Example: x² - 4 = (x - 2)(x + 2).

If you need to review any of these concepts, consult your algebra and pre-calculus textbooks or online resources such as Khan Academy. A solid foundation in these areas is essential for success in understanding limits and continuity.

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## 4. MAIN CONTENT

### 4.1 The Intuitive Idea of a Limit

Overview: The concept of a limit is fundamental to calculus. It describes the value that a function "approaches" as the input approaches a particular value. This is not necessarily the actual value of the function at that point, but rather what the function seems to be heading towards.

The Core Concept: Imagine walking along a path towards a destination. The limit is like predicting where you will end up, even if you never actually reach that exact spot. Formally, the limit of a function f(x) as x approaches a is L, written as lim (x→a) f(x) = L, if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a (but not equal to a).

This definition is crucial. It emphasizes "approaching" rather than "equaling." The function doesn't need to be defined at x = a for the limit to exist. It only matters what happens as x gets closer and closer to a from both sides (from values less than a and from values greater than a). If the function approaches different values from the left and right, the limit does not exist. This is a two-sided limit, and both sides must agree.

Think of a function as a machine that takes an input x and produces an output f(x). When we're talking about a limit, we're interested in what happens to the output of the machine as we feed it inputs that are getting closer and closer to a specific value. If the outputs get closer and closer to a specific value L, then L is the limit of the function as x approaches a.

Concrete Examples:

Example 1: Consider the function f(x) = x + 2. What is the limit of f(x) as x approaches 3?

Setup: We want to find lim (x→3) (x + 2).
Process: As x gets closer to 3 (e.g., 2.9, 2.99, 2.999, or 3.1, 3.01, 3.001), the value of x + 2 gets closer to 5.
Result: lim (x→3) (x + 2) = 5. In this case, the function is defined at x = 3, and f(3) = 5. The limit is simply the value of the function at that point.
Why this matters: This simple example illustrates the basic idea of a limit. It shows how we can determine the value a function approaches as its input gets closer to a specific value.

Example 2: Consider the function g(x) = (x² - 1) / (x - 1). What is the limit of g(x) as x approaches 1?

Setup: We want to find lim (x→1) (x² - 1) / (x - 1).
Process: Notice that g(1) is undefined because we would be dividing by zero. However, we can simplify the expression: (x² - 1) / (x - 1) = (x - 1)(x + 1) / (x - 1) = x + 1 (for x ≠ 1). As x gets closer to 1, x + 1 gets closer to 2.
Result: lim (x→1) (x² - 1) / (x - 1) = 2. Even though g(1) is undefined, the limit exists and is equal to 2.
Why this matters: This example demonstrates that a limit can exist even when the function is not defined at the point in question. This is a crucial concept in calculus.

Analogies & Mental Models:

Think of it like... aiming a dart at a bullseye. The limit is the bullseye itself. You might not hit the bullseye every time, but the closer you aim, the closer your dart will be to the bullseye. The function's value is where the dart actually lands, while the limit is the bullseye you're aiming for.
Where the analogy breaks down: The dart analogy is imperfect because it focuses on a single attempt. Limits are about what always happens as you get closer and closer.

Common Misconceptions:

❌ Students often think that the limit of a function at a point is simply the value of the function at that point.
✓ Actually, the limit is the value the function approaches as the input gets arbitrarily close to the point, regardless of whether the function is defined at that point.
Why this confusion happens: In many simple cases, the limit is equal to the function's value. However, it's important to remember that this is not always the case, especially when dealing with functions that have discontinuities or are undefined at certain points.

Visual Description:

Imagine a graph of a function. As you trace the graph from both the left and the right, getting closer and closer to a specific x-value (let's call it 'a'), observe what y-value the graph seems to be approaching. That y-value is the limit of the function as x approaches 'a'. If the graph approaches different y-values from the left and right, then the limit does not exist. You can visualize this by drawing a graph with a "jump" in it.

Practice Check:

What is the limit of h(x) = 2x - 1 as x approaches 4?

Answer: lim (x→4) (2x - 1) = 2(4) - 1 = 7.

Connection to Other Sections:

This section provides the foundational understanding of what a limit is. The next sections will build upon this by introducing techniques for calculating limits and exploring the concept of continuity, which is directly related to the existence of limits.

### 4.2 Formal Definition of a Limit (ε-δ Definition)

Overview: The informal definition of a limit is helpful for understanding the concept, but it lacks the rigor needed for mathematical proofs. The epsilon-delta (ε-δ) definition provides a precise and unambiguous way to define a limit.

The Core Concept: The ε-δ definition of a limit states that for every ε > 0 (epsilon, representing a small positive number), there exists a δ > 0 (delta, also a small positive number) such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Let's break this down:

ε > 0: Epsilon represents an arbitrarily small distance around the limit L. Think of it as a tolerance level. We want the function's output f(x) to be within this tolerance of L.
δ > 0: Delta represents a distance around the input value a. We need to find a delta such that if x is within this distance of a (but not equal to a), then f(x) will be within the epsilon distance of L.
0 < |x - a| < δ: This means that x is within a distance of δ from a, but x is not equal to a. The "0 <" part ensures that we're not considering the value of the function at x = a, only its behavior near a.
|f(x) - L| < ε: This means that the distance between f(x) and L is less than ε. In other words, f(x) is within the tolerance level of L.

The ε-δ definition is essentially a game. Someone gives you an ε (a tolerance), and you have to find a δ (a distance around a) that guarantees that f(x) will be within that tolerance of L. If you can always find such a δ, no matter how small the ε is, then the limit of f(x) as x approaches a is indeed L.

Concrete Examples:

Example 1: Prove that lim (x→2) (3x - 2) = 4 using the ε-δ definition.

Setup: We need to show that for every ε > 0, there exists a δ > 0 such that if 0 < |x - 2| < δ, then |(3x - 2) - 4| < ε.
Process: Let's simplify the expression |(3x - 2) - 4|: |(3x - 2) - 4| = |3x - 6| = 3|x - 2|. We want this to be less than ε, so we have 3|x - 2| < ε. Dividing both sides by 3, we get |x - 2| < ε/3. Therefore, we can choose δ = ε/3.
Result: If we choose δ = ε/3, then whenever 0 < |x - 2| < δ, we have |(3x - 2) - 4| = 3|x - 2| < 3(ε/3) = ε. This proves that lim (x→2) (3x - 2) = 4.
Why this matters: This example demonstrates how to use the ε-δ definition to rigorously prove the existence of a limit for a linear function.

Example 2: Prove that lim (x→1) x² = 1 using the ε-δ definition.

Setup: We need to show that for every ε > 0, there exists a δ > 0 such that if 0 < |x - 1| < δ, then |x² - 1| < ε.
Process: We can factor |x² - 1| as |(x - 1)(x + 1)| = |x - 1||x + 1|. We want to find a δ such that |x - 1||x + 1| < ε. The tricky part here is the |x + 1| term. We need to bound it. Let's assume that δ ≤ 1. Then, if |x - 1| < δ, we have |x - 1| < 1, which implies -1 < x - 1 < 1, so 0 < x < 2. Therefore, |x + 1| < 3. Now we have |x - 1||x + 1| < 3|x - 1|. We want 3|x - 1| < ε, so |x - 1| < ε/3. To satisfy both δ ≤ 1 and |x - 1| < ε/3, we choose δ = min(1, ε/3).
Result: If we choose δ = min(1, ε/3), then whenever 0 < |x - 1| < δ, we have |x² - 1| = |x - 1||x + 1| < (ε/3) 3 = ε. This proves that lim (x→1) x² = 1.
Why this matters: This example demonstrates how to use the ε-δ definition for a quadratic function. It also highlights the need to bound terms to make the proof work.

Analogies & Mental Models:

Think of it like... a game of precision. Epsilon is the target area, and delta is the aiming circle. You want to make your aiming circle small enough so that every shot you take within that circle lands within the target area.
Where the analogy breaks down: The game analogy is a one-time event. The ε-δ definition requires that you can always find a suitable delta for any given epsilon.

Common Misconceptions:

❌ Students often think that they need to find a specific value for delta.
✓ Actually, you only need to show that a delta exists that satisfies the condition. You can usually express delta in terms of epsilon.
Why this confusion happens: The wording of the definition can be confusing. It's important to understand that the goal is to demonstrate the existence of a suitable delta, not to find a unique value.

Visual Description:

Draw a graph of a function f(x). Pick a point a on the x-axis and a potential limit L on the y-axis. Draw a horizontal band of width 2ε centered around L. The ε-δ definition says that you should be able to find a vertical band of width 2δ centered around a such that whenever x is within the vertical band (but not equal to a), the corresponding f(x) value is within the horizontal band. If you can do this for any ε, then the limit exists and is equal to L.

Practice Check:

Explain in your own words what the ε-δ definition of a limit means.

Answer: The ε-δ definition states that the limit of a function f(x) as x approaches a is L if, for any desired level of closeness (ε) to L, we can find a region around a (defined by δ) such that all values of f(x) for x in that region (excluding a itself) are within the desired level of closeness to L.

Connection to Other Sections:

While the ε-δ definition can seem abstract, it provides the necessary rigor for proving limit properties and theorems. The next section will explore techniques for calculating limits without directly using the ε-δ definition, but these techniques are ultimately justified by the ε-δ definition.

### 4.3 Techniques for Calculating Limits

Overview: The ε-δ definition is useful for proving that a limit exists, but it's often cumbersome to use for actually calculating limits. Fortunately, there are several techniques that make limit calculations much easier.

The Core Concept: There are several techniques for calculating limits. The most common ones include:

1. Direct Substitution: If f(x) is a continuous function at x = a, then lim (x→a) f(x) = f(a). This is the simplest technique and works for many common functions, such as polynomials, exponentials, and trigonometric functions (within their domains).
2. Factoring: If direct substitution results in an indeterminate form (e.g., 0/0), try factoring the numerator and denominator and canceling common factors. This technique is often useful for rational functions.
3. Rationalizing: If the function involves radicals, try rationalizing the numerator or denominator to simplify the expression. This often involves multiplying by the conjugate.
4. Limit Laws: Several limit laws allow you to break down complex limits into simpler ones. These laws include the sum law, difference law, product law, quotient law (provided the limit of the denominator is not zero), and power law.

Concrete Examples:

Example 1: Direct Substitution: Find lim (x→2) (x³ + 2x - 1).

Setup: We want to find lim (x→2) (x³ + 2x - 1).
Process: Since f(x) = x³ + 2x - 1 is a polynomial, it's continuous everywhere. Therefore, we can use direct substitution: f(2) = (2)³ + 2(2) - 1 = 8 + 4 - 1 = 11.
Result: lim (x→2) (x³ + 2x - 1) = 11.
Why this matters: This demonstrates the simplest and most straightforward technique for finding limits.

Example 2: Factoring: Find lim (x→-3) (x² + x - 6) / (x + 3).

Setup: We want to find lim (x→-3) (x² + x - 6) / (x + 3).
Process: Direct substitution results in 0/0, which is an indeterminate form. We can factor the numerator: (x² + x - 6) = (x + 3)(x - 2). Therefore, (x² + x - 6) / (x + 3) = (x + 3)(x - 2) / (x + 3) = x - 2 (for x ≠ -3). Now we can use direct substitution: lim (x→-3) (x - 2) = -3 - 2 = -5.
Result: lim (x→-3) (x² + x - 6) / (x + 3) = -5.
Why this matters: This example shows how factoring can be used to simplify a rational function and eliminate the indeterminate form.

Example 3: Rationalizing: Find lim (x→0) (√(x + 1) - 1) / x.

Setup: We want to find lim (x→0) (√(x + 1) - 1) / x.
Process: Direct substitution results in 0/0. We can rationalize the numerator by multiplying by the conjugate: ((√(x + 1) - 1) / x) ((√(x + 1) + 1) / (√(x + 1) + 1)) = ((x + 1) - 1) / (x(√(x + 1) + 1)) = x / (x(√(x + 1) + 1)) = 1 / (√(x + 1) + 1) (for x ≠ 0). Now we can use direct substitution: lim (x→0) 1 / (√(x + 1) + 1) = 1 / (√(0 + 1) + 1) = 1 / (1 + 1) = 1/2.
Result: lim (x→0) (√(x + 1) - 1) / x = 1/2.
Why this matters: This illustrates how rationalizing can be used to simplify expressions involving radicals and eliminate the indeterminate form.

Example 4: Limit Laws: Given lim (x→a) f(x) = 3 and lim (x→a) g(x) = -2, find lim (x→a) (2f(x) + 3g(x)).

Setup: We want to find lim (x→a) (2f(x) + 3g(x)).
Process: Using the limit laws, we can break this down: lim (x→a) (2f(x) + 3g(x)) = 2 lim (x→a) f(x) + 3 lim (x→a) g(x) = 2 3 + 3 (-2) = 6 - 6 = 0.
Result: lim (x→a) (2f(x) + 3g(x)) = 0.
Why this matters: This shows how limit laws can be used to simplify complex limits by breaking them down into simpler ones.

Analogies & Mental Models:

Think of it like... a toolbox with different tools for different tasks. Direct substitution is like a screwdriver – it works well for simple tasks. Factoring and rationalizing are like wrenches and pliers – they're needed for more complex tasks. Limit laws are like a set of instructions that tell you how to combine the tools to solve a problem.
Where the analogy breaks down: The toolbox analogy suggests that you choose one tool at a time. Sometimes, you need to use multiple techniques in combination to find a limit.

Common Misconceptions:

❌ Students often try to use direct substitution even when it results in an indeterminate form.
✓ Actually, direct substitution only works if the function is continuous at the point in question. If you get an indeterminate form, you need to use a different technique.
Why this confusion happens: Direct substitution is the simplest technique, so students often try to apply it in all cases.

Visual Description:

When factoring, visualize canceling out a common factor in the numerator and denominator of a rational function. This visually "removes" the source of the indeterminate form. When rationalizing, visualize multiplying by the conjugate as a way to "eliminate" the radical from the numerator or denominator.

Practice Check:

Find lim (x→4) (x² - 16) / (x - 4).

Answer: Factoring the numerator, we get (x + 4)(x - 4) / (x - 4) = x + 4 (for x ≠ 4). Then, lim (x→4) (x + 4) = 8.

Connection to Other Sections:

This section provides the practical tools for calculating limits. The next section will explore one-sided limits, which are important for understanding the behavior of functions near points where they may not be defined or may have discontinuities.

### 4.4 One-Sided Limits

Overview: Sometimes, the behavior of a function as it approaches a point from the left is different from its behavior as it approaches from the right. In these cases, we need to consider one-sided limits.

The Core Concept: A one-sided limit describes the behavior of a function as it approaches a point from either the left or the right.

Left-Hand Limit: The limit of f(x) as x approaches a from the left is denoted as lim (x→a^-) f(x) = L. This means that as x gets closer to a from values less than a, f(x) gets closer to L.
Right-Hand Limit: The limit of f(x) as x approaches a from the right is denoted as lim (x→a⁺) f(x) = L. This means that as x gets closer to a from values greater than a, f(x) gets closer to L.

A two-sided limit, lim (x→a) f(x), exists and is equal to L if and only if both the left-hand limit and the right-hand limit exist and are equal to L. That is, lim (x→a) f(x) = L if and only if lim (x→a^-) f(x) = L and lim (x→a⁺) f(x) = L.

Concrete Examples:

Example 1: Consider the piecewise function:

f(x) = x + 1, if x < 2
f(x) = 3, if x ≥ 2

Find the left-hand and right-hand limits as x approaches 2.

Setup: We want to find lim (x→2^-) f(x) and lim (x→2⁺) f(x).
Process: As x approaches 2 from the left (x < 2), we use the first part of the function: f(x) = x + 1. Therefore, lim (x→2^-) f(x) = lim (x→2^-) (x + 1) = 2 + 1 = 3. As x approaches 2 from the right (x ≥ 2), we use the second part of the function: f(x) = 3. Therefore, lim (x→2⁺) f(x) = lim (x→2⁺) 3 = 3.
Result: lim (x→2^-) f(x) = 3 and lim (x→2⁺) f(x) = 3. Since both one-sided limits are equal to 3, the two-sided limit exists and is equal to 3.
Why this matters: This example shows how to calculate one-sided limits for a piecewise function.

Example 2: Consider the piecewise function:

g(x) = x², if x < 1
g(x) = 2x, if x ≥ 1

Find the left-hand and right-hand limits as x approaches 1.

Setup: We want to find lim (x→1^-) g(x) and lim (x→1⁺) g(x).
Process: As x approaches 1 from the left (x < 1), we use the first part of the function: g(x) = x². Therefore, lim (x→1^-) g(x) = lim (x→1^-) x² = 1² = 1. As x approaches 1 from the right (x ≥ 1), we use the second part of the function: g(x) = 2x. Therefore, lim (x→1⁺) g(x) = lim (x→1⁺) 2x = 2(1) = 2.
Result: lim (x→1^-) g(x) = 1 and lim (x→1⁺) g(x) = 2. Since the left-hand limit and the right-hand limit are not equal, the two-sided limit does not exist.
Why this matters: This example demonstrates a case where the one-sided limits exist but are not equal, which means that the two-sided limit does not exist. This is a key concept for understanding continuity.

Analogies & Mental Models:

Think of it like... approaching a bridge from two different directions. If the bridge is continuous (smooth), you'll end up at the same point regardless of which direction you approach from. But if the bridge has a gap in the middle, you'll end up at different points depending on which direction you approach from.
Where the analogy breaks down: The bridge analogy is static. One-sided limits are about the approach, not necessarily the final destination.

Common Misconceptions:

❌ Students often think that if a function is defined at a point, then the limit must exist at that point.
✓ Actually, the existence of a limit depends on the behavior of the function near the point, not necessarily at the point.
Why this confusion happens: It's easy to confuse the value of the function at a point with the limit of the function as it approaches that point.

Visual Description:

Draw a graph of a piecewise function that has a "jump" at a certain x-value. As you trace the graph from the left, you'll approach one y-value. As you trace the graph from the right, you'll approach a different y-value. This visually demonstrates that the one-sided limits are different, and therefore the two-sided limit does not exist.

Practice Check:

Consider the function h(x) = |x| / x. What are the left-hand and right-hand limits as x approaches 0? Does the two-sided limit exist?

Answer: h(x) = -1 for x < 0 and h(x) = 1 for x > 0. Therefore, lim (x→0^-) h(x) = -1 and lim (x→0⁺) h(x) = 1. Since the one-sided limits are not equal, the two-sided limit does not exist.

Connection to Other Sections:

One-sided limits are essential for understanding continuity. A function is continuous at a point only if the limit exists at that point, which means that the one-sided limits must exist and be equal.

### 4.5 Definition of Continuity

Overview: Continuity is a fundamental concept in calculus that describes functions that have no abrupt jumps, breaks, or holes.

The Core Concept: A function f(x) is continuous at a point x = a if the following three conditions are met:

1. f(a) is defined (i.e., a is in the domain of f).
2. lim (x→a) f(x) exists (i.e., both the left-hand and right-hand limits exist and are equal).
3. lim (x→a) f(x) = f(a) (i.e., the limit of f(x) as x approaches a is equal to the value of the function at a).

If any of these conditions are not met, then the function is said to be discontinuous at x = a.

A function is continuous on an interval if it is continuous at every point in the interval.

Concrete Examples:

Example 1: Is the function f(x) = x² + 2x - 1 continuous at x = 3?

Setup: We need to check the three conditions for continuity.
Process:
1. f(3)* = (3)² +

Okay, here is a comprehensive lesson plan on Calculus: Limits and Continuity, designed to be highly detailed, structured, and engaging for high school students (grades 9-12). I've aimed for depth, clarity, and real-world relevance throughout.

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## 1. INTRODUCTION

### 1.1 Hook & Context

Imagine you're designing a roller coaster. You want the ride to be thrilling, but also safe. You need to ensure the track is continuous – no sudden drops or jumps that could derail the coaster. To achieve this, you need to understand how functions behave as they approach certain points, ensuring smooth transitions. Now, think about creating a self-driving car. The car needs to react instantly to changes in its environment – speed, distance from other cars, traffic lights. It needs to predict where objects will be an instant from now, not just where they are now. This requires understanding how functions change over infinitesimally small intervals of time, which is the core idea behind limits.

Have you ever wondered how your phone's GPS knows your exact location, even when you're moving? Or how video games create realistic animations? Or how engineers design bridges that can withstand extreme forces? All these things rely on the power of calculus, and it all starts with understanding the fundamental concepts of limits and continuity. These concepts provide the foundation for understanding rates of change, optimization, and accumulation – the core ideas of calculus. They are the building blocks for solving complex problems in science, engineering, economics, and many other fields.

### 1.2 Why This Matters

Understanding limits and continuity is not just about passing a math test; it's about developing a powerful way of thinking about the world. In the real world, very few things are perfectly still or constant. Everything is in motion, everything is changing, and limits and continuity provide the tools to analyze and understand that change. They form the basis for understanding derivatives (rates of change) and integrals (accumulation), which are essential for modeling and solving problems in almost every scientific and engineering discipline.

For example, physicists use calculus to describe the motion of objects, from planets to subatomic particles. Engineers use it to design structures, optimize processes, and control systems. Economists use it to model market behavior and predict economic trends. Computer scientists use it to develop algorithms and create realistic simulations. Even fields like medicine and biology rely on calculus for analyzing population growth, drug dosages, and disease spread.

This lesson builds directly on your prior knowledge of functions, graphs, and algebra. It's a natural extension of these concepts, allowing you to move from static descriptions to dynamic analyses. Mastering limits and continuity will pave the way for you to tackle more advanced topics in calculus, such as differentiation and integration, and will open doors to a wide range of exciting career paths in STEM fields.

### 1.3 Learning Journey Preview

In this lesson, we'll embark on a journey to explore the fascinating world of limits and continuity. We'll start by defining what a limit is and how to calculate it, both graphically and algebraically. We'll then delve into the concept of continuity and explore different types of discontinuities. We'll learn how to determine whether a function is continuous at a point and over an interval. We'll also examine the properties of continuous functions and their applications in solving real-world problems. Finally, we'll connect these concepts to future topics in calculus, such as derivatives and integrals, demonstrating how they form the foundation for more advanced mathematical ideas. We'll see how these concepts are applied in various fields, and learn about the careers that utilize them. By the end of this lesson, you'll have a solid understanding of limits and continuity and be well-prepared for further exploration of calculus.

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## 2. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

Define the concept of a limit and explain its intuitive meaning.
Calculate limits of functions graphically, numerically, and algebraically.
Identify and classify different types of discontinuities (removable, jump, infinite).
Determine whether a function is continuous at a point using the three-part definition of continuity.
Apply the Intermediate Value Theorem to determine the existence of solutions to equations.
Evaluate limits involving infinity and analyze the asymptotic behavior of functions.
Explain the relationship between limits and continuity, and how they form the foundation for calculus.
Analyze real-world scenarios involving limits and continuity, such as rates of change and optimization problems.

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## 3. PREREQUISITE KNOWLEDGE

Before diving into limits and continuity, you should have a solid understanding of the following concepts:

Functions: Definition, notation (f(x)), domain, range, and different types of functions (linear, quadratic, polynomial, rational, trigonometric).
Graphs of Functions: Ability to graph functions accurately, identify key features such as intercepts, slopes, and asymptotes.
Algebraic Manipulation: Proficiency in simplifying expressions, solving equations, and working with inequalities.
Coordinate Geometry: Understanding of the Cartesian coordinate system, distance formula, and equation of a line.
Interval Notation: Ability to express sets of numbers using interval notation.

Quick Review:

Function: A relation between a set of inputs (domain) and a set of permissible outputs (range) with the property that each input is related to exactly one output.
Graph of a Function: A visual representation of the function on a coordinate plane, where the x-axis represents the input values (domain) and the y-axis represents the output values (range).
Algebraic Manipulation: The process of rewriting an expression or equation using algebraic rules and properties to simplify it or solve for an unknown variable.

Where to Review if Needed:

Khan Academy: Functions, Graphs, Algebra
Your previous algebra and pre-calculus textbooks.

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## 4. MAIN CONTENT

### 4.1 Introduction to Limits

Overview: The concept of a limit is fundamental to calculus. It describes the behavior of a function as its input approaches a particular value. It tells us what value a function "approaches" as the input gets arbitrarily close to a certain point, without necessarily reaching that point.

The Core Concept: Imagine you're walking towards a door. As you get closer and closer to the door, your distance from the door gets smaller and smaller. The limit is like asking: what is your distance approaching as you get infinitely close to the door? The answer is zero. However, you can get infinitely close without actually touching the door. This is the essence of a limit.

Formally, we say that the limit of a function f(x) as x approaches a value 'c' is 'L', written as lim (x→c) f(x) = L, if the values of f(x) get arbitrarily close to L as x gets arbitrarily close to c, but not necessarily equal to c. This means that we can make f(x) as close to L as we want by choosing x sufficiently close to c. The key is that we're interested in the behavior of the function near c, not necessarily at c. In fact, f(c) may not even be defined!

It's important to distinguish between the value of a function at a point, f(c), and the limit of the function as x approaches that point, lim (x→c) f(x). The value f(c) is simply the output of the function when x is equal to c. The limit, on the other hand, describes the function's behavior as x gets close to c, regardless of what the function actually does at c. They are only equal when the function is continuous at that point (more on this later).

Concrete Examples:

Example 1: Consider the function f(x) = x + 2. Let's find the limit as x approaches 3.
Setup: We want to find lim (x→3) (x + 2).
Process: As x gets closer and closer to 3 (e.g., 2.9, 2.99, 2.999, 3.1, 3.01, 3.001), the value of x + 2 gets closer and closer to 5 (e.g., 4.9, 4.99, 4.999, 5.1, 5.01, 5.001).
Result: Therefore, lim (x→3) (x + 2) = 5. In this simple case, we can see that the limit is equal to the value of the function at x = 3, f(3) = 3 + 2 = 5.
Why this matters: This example illustrates the basic idea of a limit: as x approaches a certain value, the function approaches a certain value.

Example 2: Consider the function g(x) = (x^2 - 1) / (x - 1). Notice that g(1) is undefined because it would result in division by zero. However, we can still find the limit as x approaches 1.
Setup: We want to find lim (x→1) (x^2 - 1) / (x - 1).
Process: We can factor the numerator: (x^2 - 1) = (x - 1)(x + 1). So, g(x) = (x - 1)(x + 1) / (x - 1). For x ≠ 1, we can cancel the (x - 1) terms, giving us g(x) = x + 1. As x gets closer and closer to 1 (but not equal to 1), x + 1 gets closer and closer to 2.
Result: Therefore, lim (x→1) (x^2 - 1) / (x - 1) = 2. Even though the function is undefined at x = 1, the limit exists and is equal to 2.
Why this matters: This example highlights that the limit of a function at a point doesn't necessarily depend on the value of the function at that point. The limit describes the function's behavior near the point.

Analogies & Mental Models:

Think of it like approaching a target with a dart. The limit is the target itself. You can throw darts closer and closer to the target, but you don't necessarily have to hit it to know where the target is.
Think of it like a car approaching a stop sign. The limit is the position of the stop sign. The car can slow down and get arbitrarily close to the stop sign, but it doesn't necessarily have to come to a complete stop at the sign to know where the stop sign is located.
Where the analogy breaks down: These analogies are helpful for visualizing the concept of approaching a value, but they don't capture the full nuance of limits. Limits can be more complex than simply approaching a physical object. For example, limits can involve functions that oscillate wildly or have infinite values.

Common Misconceptions:

❌ Students often think that the limit of a function at a point is simply the value of the function at that point.
✓ Actually, the limit describes the behavior of the function near the point, not necessarily at the point. The function may not even be defined at that point!
Why this confusion happens: Students tend to overgeneralize from simple examples where the limit is equal to the function value. It's crucial to emphasize examples where the function is undefined or discontinuous at the point in question.

Visual Description:

Imagine a graph of a function. As you trace the graph from both the left and the right towards a particular x-value (c), observe what y-value the graph is approaching. If the graph approaches the same y-value (L) from both sides, then the limit as x approaches c is L. If the graph approaches different y-values from the left and right, or if the graph oscillates wildly, then the limit does not exist.

Practice Check:

What is the limit of f(x) = 2x - 1 as x approaches 4?

Answer: lim (x→4) (2x - 1) = 2(4) - 1 = 7. As x gets closer to 4, 2x-1 gets closer to 7.

Connection to Other Sections:

This section introduces the fundamental concept of a limit, which is essential for understanding continuity (Section 4.2) and derivatives (future topic). The ability to calculate limits is a prerequisite for determining whether a function is continuous and for calculating derivatives.

### 4.2 Continuity

Overview: Continuity is a property of functions that describes whether the function's graph can be drawn without lifting your pen from the paper. A continuous function has no breaks, jumps, or holes in its graph.

The Core Concept: Intuitively, a function is continuous at a point if you can draw its graph through that point without lifting your pen. More formally, a function f(x) is continuous at a point x = c if the following three conditions are met:

1. f(c) exists: The function must be defined at x = c. There can't be a hole in the graph at that point.
2. lim (x→c) f(x) exists: The limit of the function as x approaches c must exist. This means the function must approach the same value from both the left and the right.
3. lim (x→c) f(x) = f(c): The limit of the function as x approaches c must be equal to the value of the function at x = c. In other words, the function must approach the value it actually has at that point.

If any of these three conditions are not met, then the function is said to be discontinuous at x = c. There are three main types of discontinuities:

Removable Discontinuity: A discontinuity where the limit exists, but the function is either undefined at that point or the value of the function at that point is not equal to the limit. This type of discontinuity can be "removed" by redefining the function at that point to be equal to the limit. This often appears as a "hole" in the graph.
Jump Discontinuity: A discontinuity where the limit from the left and the limit from the right both exist, but they are not equal. This type of discontinuity creates a "jump" in the graph.
Infinite Discontinuity: A discontinuity where the function approaches infinity (or negative infinity) as x approaches c. This type of discontinuity often occurs at vertical asymptotes.

A function is said to be continuous on an interval if it is continuous at every point in that interval.

Concrete Examples:

Example 1: Consider the function f(x) = x^2. Let's check if it's continuous at x = 2.
Setup: We need to verify the three conditions of continuity at x = 2.
Process:
1. f(2) = 2^2 = 4 (exists)
2. lim (x→2) x^2 = 4 (exists)
3. lim (x→2) x^2 = f(2) = 4
Result: Since all three conditions are met, f(x) = x^2 is continuous at x = 2. In fact, it is continuous for all real numbers.
Why this matters: This example shows a simple case of a continuous function. Polynomial functions like x^2 are always continuous everywhere.

Example 2: Consider the function h(x) = { x + 1, if x < 1; 3, if x = 1; x^2, if x > 1 }. Let's check if it's continuous at x = 1.
Setup: We need to verify the three conditions of continuity at x = 1.
Process:
1. h(1) = 3 (exists)
2. lim (x→1-) h(x) = lim (x→1-) (x + 1) = 2 (limit from the left)
3. lim (x→1+) h(x) = lim (x→1+) (x^2) = 1 (limit from the right)
Since the limit from the left (2) is not equal to the limit from the right (1), the limit as x approaches 1 does not exist.
Result: Since the limit does not exist, h(x) is discontinuous at x = 1. This is a jump discontinuity.
Why this matters: This example shows a discontinuous function. The jump discontinuity occurs because the function "jumps" from one value to another at x = 1.

Analogies & Mental Models:

Think of it like a road. A continuous road is smooth and has no breaks or potholes. A discontinuous road has breaks, potholes, or sudden jumps.
Think of it like a smoothly flowing river. A continuous river flows smoothly without any sudden changes in direction or water level. A discontinuous river has waterfalls or sudden changes in width.
Where the analogy breaks down: These analogies are helpful for visualizing the concept of continuity, but they don't capture the full mathematical rigor. Continuity is a precise mathematical definition, and it's important to understand the formal conditions.

Common Misconceptions:

❌ Students often think that if a function is defined at a point, then it must be continuous at that point.
✓ Actually, a function can be defined at a point but still be discontinuous there. The limit must exist and be equal to the function value for the function to be continuous.
Why this confusion happens: Students may not fully understand the difference between the value of a function at a point and the limit of the function as x approaches that point.

Visual Description:

A continuous function can be drawn without lifting your pen. A removable discontinuity will appear as a hole in the graph. A jump discontinuity will appear as a sudden jump in the graph. An infinite discontinuity will appear as a vertical asymptote.

Practice Check:

Is the function f(x) = 1/x continuous at x = 0? Why or why not?

Answer: No, f(x) = 1/x is not continuous at x = 0. f(0) is undefined, so the first condition of continuity is not met. Also, lim (x→0) 1/x does not exist (it approaches infinity). This is an infinite discontinuity.

Connection to Other Sections:

This section builds on the concept of limits (Section 4.1). The definition of continuity relies heavily on the concept of a limit. Understanding continuity is crucial for understanding the Intermediate Value Theorem (Section 4.3) and for future topics in calculus, such as differentiability.

### 4.3 The Intermediate Value Theorem (IVT)

Overview: The Intermediate Value Theorem (IVT) is a powerful theorem that guarantees the existence of a value within a given interval, under certain conditions.

The Core Concept: The IVT states that if f(x) is a continuous function on a closed interval [a, b], and k is any number between f(a) and f(b), then there exists at least one number c in the interval (a, b) such that f(c) = k. In simpler terms, if a continuous function takes on two values, it must take on every value in between.

Think of it this way: Imagine you're hiking up a mountain. If you start at an elevation of 1000 feet and end at an elevation of 5000 feet, then at some point during your hike, you must have been at every elevation between 1000 and 5000 feet. This is essentially what the IVT says.

The IVT is particularly useful for proving the existence of solutions to equations. If we can find two values, 'a' and 'b', such that f(a) and f(b) have opposite signs, then the IVT guarantees that there exists at least one value 'c' between 'a' and 'b' such that f(c) = 0. In other words, there is at least one root (x-intercept) of the function in that interval.

Concrete Examples:

Example 1: Show that the equation x^3 - 4x + 2 = 0 has a solution between 1 and 2.
Setup: Let f(x) = x^3 - 4x + 2. We want to show that there exists a value 'c' between 1 and 2 such that f(c) = 0.
Process:
1. Check if f(x) is continuous on the interval [1, 2]. Since f(x) is a polynomial, it is continuous everywhere.
2. Evaluate f(1) and f(2):
f(1) = 1^3 - 4(1) + 2 = -1
f(2) = 2^3 - 4(2) + 2 = 2
Since f(1) = -1 and f(2) = 2, and 0 is between -1 and 2, the IVT applies.
Result: By the IVT, there exists at least one value 'c' in the interval (1, 2) such that f(c) = 0. Therefore, the equation x^3 - 4x + 2 = 0 has a solution between 1 and 2.
Why this matters: This example demonstrates how the IVT can be used to prove the existence of solutions to equations without actually finding the solutions.

Example 2: Does the equation cos(x) = x have a solution?
Setup: Let f(x) = cos(x) - x. We want to show that there exists a value 'c' such that f(c) = 0.
Process:
1. Check if f(x) is continuous. Both cos(x) and x are continuous, so their difference is also continuous.
2. Choose two values, 'a' and 'b', and evaluate f(a) and f(b):
Let a = 0: f(0) = cos(0) - 0 = 1
Let b = π/2: f(π/2) = cos(π/2) - π/2 = 0 - π/2 = -π/2
Since f(0) = 1 and f(π/2) = -π/2, and 0 is between 1 and -π/2, the IVT applies.
Result: By the IVT, there exists at least one value 'c' in the interval (0, π/2) such that f(c) = 0. Therefore, the equation cos(x) = x has a solution.
Why this matters: This example shows how the IVT can be used to prove the existence of solutions to transcendental equations, which can be difficult or impossible to solve algebraically.

Analogies & Mental Models:

Think of it like a rope stretched between two points at different heights. If the rope is continuous (no breaks), then it must cross every height between the two endpoints.
Think of it like a temperature reading. If the temperature at noon is 20°C and the temperature at 6 pm is 10°C, and the temperature changes continuously throughout the day, then at some point during the afternoon, the temperature must have been every value between 10°C and 20°C.
Where the analogy breaks down: These analogies are helpful for visualizing the IVT, but they don't capture the full mathematical rigor. The IVT only applies to continuous functions, and it only guarantees the existence of a value, not the uniqueness of the value.

Common Misconceptions:

❌ Students often think that the IVT can be used to find the value 'c' such that f(c) = k.
✓ Actually, the IVT only guarantees the existence of such a value. It doesn't provide a method for finding it.
Why this confusion happens: Students may misinterpret the theorem as a method for solving equations, rather than a tool for proving the existence of solutions.

Visual Description:

Imagine a continuous graph between two points (a, f(a)) and (b, f(b)). Draw a horizontal line at a y-value 'k' between f(a) and f(b). The IVT guarantees that the graph must intersect this horizontal line at least once between x = a and x = b.

Practice Check:

Given that f(x) is continuous on [0, 5], f(0) = -3, and f(5) = 2, does there exist a value 'c' between 0 and 5 such that f(c) = 0?

Answer: Yes, by the IVT. Since f(0) is negative and f(5) is positive, and 0 is between -3 and 2, there must be a value 'c' between 0 and 5 such that f(c) = 0.

Connection to Other Sections:

This section builds on the concept of continuity (Section 4.2). The IVT only applies to continuous functions. It provides a powerful application of continuity in proving the existence of solutions to equations.

### 4.4 Limits Involving Infinity

Overview: Limits involving infinity describe the behavior of a function as its input approaches infinity (or negative infinity), or as the function's output becomes infinitely large.

The Core Concept: There are two main types of limits involving infinity:

1. Limits as x approaches infinity (or negative infinity): This type of limit describes the long-term behavior of a function. We write lim (x→∞) f(x) = L if f(x) approaches the value L as x becomes infinitely large. Similarly, lim (x→-∞) f(x) = L if f(x) approaches L as x becomes infinitely small (negative). If the limit exists, the function has a horizontal asymptote at y = L.

2. Infinite Limits: This type of limit describes the behavior of a function as its output becomes infinitely large (or infinitely small). We write lim (x→c) f(x) = ∞ if f(x) becomes infinitely large as x approaches c. Similarly, lim (x→c) f(x) = -∞ if f(x) becomes infinitely small as x approaches c. If either of these limits exists, the function has a vertical asymptote at x = c.

Understanding limits involving infinity is crucial for analyzing the asymptotic behavior of functions, which is important in many applications, such as modeling population growth, radioactive decay, and electrical circuits.

Concrete Examples:

Example 1: Find lim (x→∞) (1/x).
Setup: We want to determine the behavior of the function 1/x as x becomes infinitely large.
Process: As x gets larger and larger (e.g., 10, 100, 1000, 10000), the value of 1/x gets smaller and smaller (e.g., 0.1, 0.01, 0.001, 0.0001).
Result: Therefore, lim (x→∞) (1/x) = 0. The function 1/x approaches 0 as x approaches infinity. This means the x-axis (y=0) is a horizontal asymptote of the function.
Why this matters: This example illustrates the basic idea of a limit as x approaches infinity. The function approaches a finite value as x becomes infinitely large.

Example 2: Find lim (x→0+) (1/x). (The "+" indicates we're approaching 0 from the right side).
Setup: We want to determine the behavior of the function 1/x as x approaches 0 from the right side (i.e., through positive values).
Process: As x gets closer and closer to 0 from the right (e.g., 1, 0.1, 0.01, 0.001), the value of 1/x gets larger and larger (e.g., 1, 10, 100, 1000).
Result: Therefore, lim (x→0+) (1/x) = ∞. The function 1/x approaches infinity as x approaches 0 from the right. This means the y-axis (x=0) is a vertical asymptote of the function.
Why this matters: This example illustrates the concept of an infinite limit. The function approaches infinity as x approaches a certain value.

Analogies & Mental Models:

Think of it like a runner approaching the finish line. If the runner keeps running forever, they are approaching infinity.
Think of it like a balloon being inflated. As more air is pumped into the balloon, its size approaches infinity (until it bursts).
Where the analogy breaks down: These analogies are helpful for visualizing the concept of infinity, but they don't capture the full mathematical rigor. Infinity is not a number; it's a concept that describes unbounded growth.

Common Misconceptions:

❌ Students often think that infinity is a number.
✓ Actually, infinity is not a number. It's a concept that describes unbounded growth. We can't perform arithmetic operations with infinity like we can with numbers.
Why this confusion happens: Students may have difficulty grasping the abstract concept of infinity.

Visual Description:

When evaluating lim (x→∞) f(x), look at the graph of f(x) as x moves far to the right. Does the graph approach a horizontal line? If so, the y-value of that line is the limit. When evaluating lim (x→c) f(x) = ∞, look at the graph of f(x) as x approaches 'c'. Does the graph shoot up towards positive infinity or down towards negative infinity? If so, there is a vertical asymptote at x = c.

Practice Check:

What is lim (x→∞) (3x^2 + 2x + 1) / (x^2 + 5)?

Answer: Divide both the numerator and denominator by x^2 (the highest power of x). This gives us lim (x→∞) (3 + 2/x + 1/x^2) / (1 + 5/x^2). As x approaches infinity, 2/x, 1/x^2, and 5/x^2 all approach 0. Therefore, the limit is (3 + 0 + 0) / (1 + 0) = 3.

Connection to Other Sections:

This section builds on the concept of limits (Section 4.1). It extends the idea of limits to include cases where the input or output approaches infinity. Understanding limits involving infinity is crucial for analyzing the asymptotic behavior of functions and for future topics in calculus, such as integration.

### 4.5 One-Sided Limits

Overview: One-sided limits examine the behavior of a function as it approaches a specific value from either the left or the right.

The Core Concept: A one-sided limit considers the function's behavior as x approaches a value 'c' from only one direction.

Left-Hand Limit: lim (x→c-) f(x) (The superscript "-" indicates approaching from the left or negative side) This means we are only considering values of x that are less than c. We ask: what value does f(x) approach as x gets closer and closer to c, coming only from the left?

Right-Hand Limit: lim (x→c+) f(x) (The superscript "+" indicates approaching from the right or positive side) This means we are only considering values of x that are greater than c. We ask: what value does f(x) approach as x gets closer and closer to c, coming only from the right?

For a regular (two-sided) limit to exist, both the left-hand limit and the right-hand limit must exist and be equal. That is, lim (x→c) f(x) = L if and only if lim (x→c-) f(x) = L and lim (x→c+) f(x) = L.

One-sided limits are particularly useful for analyzing functions that have different definitions on either side of a point, such as piecewise functions or functions with jump discontinuities.

Concrete Examples:

Example 1: Consider the piecewise function f(x) = { x + 1, if x < 2; x^2 - 1, if x ≥ 2 }. Find the left-hand and right-hand limits as x approaches 2.
Setup: We need to find lim (x→2-) f(x) and lim (x→2+) f(x).
Process:
For the left-hand limit, we use the definition of f(x) for x < 2: lim (x→2-) f(x) = lim (x→2-) (x + 1) = 2 + 1 = 3.
For the right-hand limit, we use the definition of f(x) for x ≥ 2: lim (x→2+) f(x) = lim (x→2+) (x^2 - 1) = 2^2 - 1 = 3.
Result: Since lim (x→2-) f(x) = 3 and lim (x→2+) f(x) = 3, the limit as x approaches 2 exists and is equal to 3: lim (x→2) f(x) = 3.
Why this matters: This example shows that even though a function is defined differently on either side of a point, the limit can still exist if the left-hand and right-hand limits are equal.

Example 2: Consider the function g(x) = { x, if x < 0; 1, if x ≥ 0 }. Find the left-hand and right-hand limits as x approaches 0.
Setup: We need to find lim (x→0-) g(x) and lim (x→0+) g(x).
Process:
For the left-hand limit, we use the definition of g(x) for x < 0: lim (x→0-) g(x) = lim (x→0-) (x) = 0.
For the right-hand limit, we use the definition of g(x) for x ≥ 0: lim (x→0+) g(x) = lim (x→0+) (1) = 1.
Result: Since lim (x→0-) g(x) = 0 and lim (x→0+) g(x) = 1, the left-hand and right