Welcome to the World of Calculus!

Welcome to Unit 1: Limits and Continuity. This is the very first step in your AP Calculus journey. If you’ve ever wondered how we can measure things that are constantly changing—like the speed of a falling ball or the growth of a population—Calculus is the answer.

In this unit, we are going to learn about Limits. Think of a limit as a way of asking, "What value is this function getting closer and closer to?" even if it never actually reaches it. It's the foundation for everything else we will do this year. Don't worry if it feels a little "abstract" at first; we'll take it one step at a time!

1.1 & 1.2: What Exactly is a Limit?

In Algebra, we usually care about what a function is at a specific point. In Calculus, we care about what a function is doing as it gets near a point.

The Limit Notation looks like this:
\( \lim_{x \to c} f(x) = L \)

This is read as: "The limit of \( f(x) \) as \( x \) approaches \( c \) equals \( L \)."

Analogy: Imagine you are walking toward a wall. The limit is the wall itself. Even if you never touch the wall, everyone can see that you are headed exactly toward it. In Calculus, the "wall" is the value \( L \).

Key Terms to Know:

Left-hand limit: Approaching from the left side (values smaller than \( c \)). Written as \( \lim_{x \to c^-} f(x) \).
Right-hand limit: Approaching from the right side (values larger than \( c \)). Written as \( \lim_{x \to c^+} f(x) \).
The General Limit: For a limit to exist, the left-hand limit and the right-hand limit must be equal. If they point to different numbers, the limit Does Not Exist (DNE).

Quick Review: The Limit Rule

If \( \lim_{x \to c^-} f(x) = L \) AND \( \lim_{x \to c^+} f(x) = L \), then \( \lim_{x \to c} f(x) = L \).
If they are different, the limit is DNE.

1.3 & 1.4: Estimating Limits from Graphs and Tables

Before we use fancy math, we can just look at pictures (graphs) or lists of numbers (tables).

Using a Graph:

To find a limit on a graph, put your fingers on the line/curve on both sides of the target \( x \)-value. Slide your fingers toward the target. Do they meet at the same height?
- If yes, that height is your limit.
- If your fingers are at different heights (like at a jump in the graph), the limit DNE.
- Common Mistake: A "hole" in the graph doesn't stop a limit from existing! The limit only cares where you are going, not what happens when you get there.

Using a Table:

If you have a table of values, look at the \( y \)-values as the \( x \)-values get closer to your target.
Example: If \( x \) is 1.9, 1.99, 1.999 and \( y \) is 3.9, 3.99, 3.999, it’s a safe bet the limit as \( x \to 2 \) is 4!

1.5 & 1.6: Algebraic Limits and Direct Substitution

Now for the "mathy" part. Most of the time, finding a limit is as easy as plugging the number in. This is called Direct Substitution.

Step 1: Plug the value \( c \) into the function.
Step 2: If you get a real number, you’re done! That’s the limit.
Step 3: If you get \( 0/0 \), this is called the Indeterminate Form. This is Calculus saying, "Keep trying! There is an answer, but it's hidden."

How to fix \( 0/0 \):

1. Factor and Cancel: See if you can factor the top or bottom to cancel out the "problem" part.
2. Rationalize: If there are square roots, multiply by the conjugate.
3. Simplify: If there are complex fractions, simplify them into a single fraction.

Did you know? The \( 0/0 \) form usually represents a "hole" in the graph. By canceling out terms, you are mathematically "filling the hole" to see what the value should have been!

1.8: The Squeeze Theorem

Don't let the name scare you! The Squeeze Theorem (also called the Sandwich Theorem) is used when a function is too messy to solve directly.

If you have a "messy" function \( g(x) \) trapped between two "nice" functions \( f(x) \) and \( h(x) \), and both nice functions are heading to the same limit \( L \), then the messy function must also go to \( L \).

Analogy: If you are walking between two friends and they both walk into a coffee shop, you are going into that coffee shop too, because you are squeezed between them!

1.9, 1.10, & 1.11: Continuity

A function is continuous if you can draw it without lifting your pencil. But in AP Calc, we need a formal definition. To be continuous at a point \( x = c \), it must pass three tests:

1. \( f(c) \) must exist (no hole or vertical asymptote at that exact spot).
2. \( \lim_{x \to c} f(x) \) must exist (the left and right sides must meet).
3. The big one: The limit must equal the function value. \( \lim_{x \to c} f(x) = f(c) \).

Types of Discontinuity:

- Removable (Hole): You have a limit, but the point is missing or in the wrong place.
- Jump: The left and right sides go to different numbers.
- Infinite: The graph goes to infinity (a vertical asymptote).

1.12 & 1.13: Limits at Infinity (Asymptotes)

Sometimes we want to know what happens to a function as \( x \) gets huge (\( \infty \)) or very small (\( -\infty \)).

Vertical Asymptotes:

If the limit as \( x \to c \) results in a number divided by zero (like \( 5/0 \)), the answer is \( \infty \) or \( -\infty \). This means there is a Vertical Asymptote at \( x = c \).

Horizontal Asymptotes:

These tell us the end behavior of the graph.
- If the denominator has a higher power (e.g., \( \frac{x}{x^2} \)), the limit is 0.
- If the powers are the same (e.g., \( \frac{3x^2}{5x^2} \)), the limit is the ratio of the coefficients: 3/5.
- If the numerator has a higher power (e.g., \( \frac{x^2}{x} \)), the limit is infinity (no horizontal asymptote).

1.14: The Intermediate Value Theorem (IVT)

The IVT is a "common sense" theorem. It says that if a function is continuous on a closed interval \( [a, b] \), it must hit every \( y \)-value between \( f(a) \) and \( f(b) \).

Analogy: If you grow from 4 feet tall to 5 feet tall, at some point in your life, you had to be exactly 4.5 feet tall. You couldn't just skip it!

Requirement: The function must be continuous for IVT to work. If there's a jump, you might skip over the value.

Summary: The "Big Ideas" of Unit 1

- Limits describe behavior near a point.
- Left = Right for a general limit to exist.
- 0/0 means you have more work to do (factor, rationalize).
- Continuity means the limit exists and equals the point's value.
- IVT guarantees that a continuous function doesn't skip any \( y \)-values.

Don't worry if this seems tricky at first! Limits are a new way of thinking. Keep practicing your algebra, and the Calculus will follow!