#AP Calculus AB/BC: Squeeze Theorem - Your Ultimate Guide 🚀

Hey there, future calculus champ! 👋 Let's dive into the Squeeze Theorem, a super useful tool for finding limits. This guide is designed to make sure you're feeling confident and ready to ace those AP Calculus questions! Let's get started!

#1.8: Determining Limits Using the Squeeze Theorem

This section focuses on using the Squeeze Theorem to determine limits when direct methods don't work. It's all about bounding a tricky function between two easier ones! Remember, this concept often appears in both multiple-choice and free-response questions. Let’s break it down!

The Squeeze Theorem (also known as the Sandwich Theorem) states that if a function is always between two other functions, and those two functions approach the same limit, then the function in the middle must also approach that same limit. Think of it like being squeezed between two friends who are walking to the same spot—you’ll end up there too! 🚶‍♂️🚶‍♀️

Formally:

If $\textcolor{red}{f(x)} \leq \textcolor{blue}{ g(x)} \leq \textcolor{green}{h(x)}$ and $\lim_{{x \to a}} f(x) = \lim_{{x \to a}} h(x) = \textcolor{orange}{L}$, then $\lim_{{x \to a}} g(x) = \textcolor{orange}{L}$.

Caption: The graph shows how g(x) is 'sandwiched' between f(x) and h(x), forcing it to approach the same limit L as x approaches a.

#📚 Background Knowledge

Before we jump into practice, make sure you're comfortable with these concepts:

• Limits: How functions behave as they approach a specific value. If you need a refresher, check out this guide.
• Basic Function Behavior: Understanding the behavior of functions like sine, cosine, and exponentials. For example, $\sin(x)$ and $\cos(x)$ are always between -1 and 1. 💡

#🧮 Squeeze Theorem Practice Problems

Let's tackle some problems to solidify your understanding! Remember, practice makes perfect. 💪

#1) Squeeze Theorem Logic

Question: Funct...