### Determining Limits Using the Squeeze Theorem: AP Calculus AB/BC Study Guide

#### Introduction

Hello, future mathematicians and limit enthusiasts! Get ready to tackle one of the coolest tools in calculus—The Squeeze Theorem! It's like the ultimate referee for functions, helping us determine limits when just eyeballing isn't enough. Let's dive into this concept and make those limits behave! 🎩📏

#### Squeeze Theorem Unveiled

Firstly, welcome back to our limits adventure. We've navigated through algebraic manipulations, dissected graphs, and squinted at tables. Now, it's time to master the Squeeze Theorem. This mighty theorem helps us figure out the limit of a function by sandwiching it between two other functions whose limits we already know. Imagine you're squeezing a watermelon seed between your fingers—it has nowhere to go but shoot out! Let’s transform that messy physics into clean calculus.

##### Background Knowledge

To wield the Squeeze Theorem effectively, brush up on the following:

**Limits:**The behavior of a function as it approaches a specific value.**Function Behavior:**A good grasp of how basic functions like sine, cosine, and exponential behave under different conditions.

##### What is the Squeeze Theorem?

The Squeeze Theorem sounds fancier than it is. Here's the deal: Suppose we have three functions ( f(x) ), ( g(x) ), and ( h(x) ) such that:

[ f(x) \leq g(x) \leq h(x) ]

for values of ( x ) close to, but not necessarily at, a point ( a ). If both ( f(x) ) and ( h(x) ) have the same limit ( L ) as ( x ) approaches ( a ), then ( g(x) ) must also approach ( L ). It’s like g(x) is playing follow-the-leader but is stuck between its two buddies on the playground.

In mathematical wizard language:

[ \text{If} \quad \lim_{{x \to a}} f(x) = \lim_{{x \to a}} h(x) = L \quad \text{then} \quad \lim_{{x \to a}} g(x) = L ]

Voilà, you have just squeezed a limit!

Here's a visual to help you get the juice:

You can see that ( g(x) ) is squeezed between ( f(x) ) and ( h(x) ), compelling ( g(x) ) to follow suit to L.

#### Squeeze Theorem in Action

Let’s practice a couple of problems to tighten our grip on the Squeeze Theorem.

##### Example 1: Squeeze Theorem Logic

Suppose ( g(x) ) and ( h(x) ) are twice-differentiable functions such that ( g(2) = h(2) = 4 ). It’s known that ( g(x) < h(x) ) for ( 1 < x < 3 ). Let ( k(x) ) be a function with:

[ g(x) \leq k(x) \leq h(x) \quad \text{for} \quad 1 < x < 3 ]

Is ( k(x) ) continuous at ( x = 2 )? Justify your answer.

**Solution:**
Since both ( g(x) ) and ( h(x) ) are twice-differentiable, they are also continuous. Thus, we know:

[ \lim_{{x \to 2}} g(x) = 4 \quad \text{and} \quad \lim_{{x \to 2}} h(x) = 4 ]

Given that ( g(x) \leq k(x) \leq h(x) ), the Squeeze Theorem can be applied, which implies:

[ \lim_{{x \to 2}} k(x) = 4 ]

Since ( k(2) ) is squeezed to 4 by ( g(2) ) and ( h(2) ):

[ k(2) = 4 ]

Thus, ( k(x) ) is continuous at ( x = 2 ) because:

[ \lim_{{x \to 2}} k(x) = k(2) = 4 ]

Boom! We've just seasoned our squeeze with continuity.

##### Example 2: Computing a Limit Using Squeeze Theorem

Find the limit of ( g(x) = x \cos \left( \frac{1}{x} \right) ) as ( x ) approaches 0 using the Squeeze Theorem.

**Solution:**

Take the fact that:

[ -1 < \cos \left( \frac{1}{x} \right) < 1 ]

Multiply everything by ( x ):

[ -x < x \cos \left( \frac{1}{x} \right) < x ]

Setting up our squeeze with ( f(x) = -x ) and ( h(x) = x ):

[ \lim_{{x \to 0}} f(x) = \lim_{{x \to 0}} -x = 0 ] [ \lim_{{x \to 0}} h(x) = \lim_{{x \to 0}} x = 0 ]

Since both limits are 0, the Squeeze Theorem tells us:

[ \lim_{{x \to 0}} x \cos \left( \frac{1}{x} \right) = 0 ]

Congratulations! Now you can squeeze limits tighter than your jeans after Thanksgiving dinner.

#### Closing Thoughts

Fantastic work! You've now mastered the Squeeze Theorem, a fundamental tool in your calculus toolkit. Prepare to see this theorem pop up in both multiple-choice questions and free-response sections of your AP exam. Stay sharp and keep squeezing those limits!

Remember, use the following steps for success:

- Identify the function for which you need to determine the limit.
- Find the functions that bound the given function.
- Ensure the limits of the bounding functions are known as ( x ) approaches the same value.

You got this! 💪📈

#### Key Terms to Review

**Approaches:**The behavior of a function as the input values get closer to a certain value.**Bounded:**A function that is limited or restricted within a certain range.**Function:**A relationship where each input value corresponds to exactly one output.**Limit:**The value a function approaches as the input gets close to a particular point.**Oscillate:**The repetitive fluctuation of a function.**Squeeze Theorem:**The process where if two functions approach the same limit, a third function squeezed between them also approaches that limit.

#### Fun Fact

Did you know that the Squeeze Theorem is also humorously known as the Sandwich Theorem? Just imagine your function getting squeezed between ( f(x) ) and ( h(x) ) like a yummy peanut butter and jelly sandwich!

Keep on squeezing those limits and let the math magic happen! 📉✨