#AP Calculus AB/BC: Limits - Your Ultimate Study Guide 🚀

Hey there, future calculus champ! This guide is designed to be your go-to resource for mastering limits, especially as you gear up for the AP exam. Let's dive in and make sure you're feeling confident and ready to ace it! 💪

#1. Understanding Limits: The Foundation

#🤔 Defining a Limit

At its core, a limit is the y-value that a function, $f(x)$, approaches as $x$ gets closer and closer to a specific value. Think of it as peeking into the function's behavior near a certain point. It's not necessarily the value at that point, but what the function is heading towards.

The notation for a limit is: $\lim\limits_{x→a}f(x) = C$, which is read as “the limit of $f(x)$ as $x$ approaches $a$ equals $C$”.

Here's a visual breakdown:

Image Courtesy of Study.com

This notation tells us that as $x$ gets closer and closer to the value $a$, the function $f(x)$ gets closer and closer to the number $C$. Remember, the limit is not equal to $C$, but rather gets closer and closer to it. 💡

#🤨 Representing Limits Numerically & Graphically

Limits can be expressed in multiple ways: numerically (with tables) and graphically (with plots).

#🔢 Representing Limits Numerically

Let’s consider the function:

$$f(x)=\frac{x^2-1}{x-1}$$

We want to find the limit as $x$ approaches 1, or $\lim...