Welcome back to AP Calculus with Fiveable! This topic focuses on determining the of a based on information given about other functions that bound it. We’ve worked through determining limits through algebraic manipulation, graphs, and tables, so let's keep building our limit skills. 🙌
⌛ Squeeze Theorem
Before we get into the nitty gritty, be sure to review some of the content we’ve already went over!
📚 Background Knowledge
To effectively use the , you should be familiar with:
Limits: Understanding how functions behave near a specific value.
Basic Function Behavior: Knowledge about how functions like sine, cosine, exponential, etc., behave for different inputs.
🧩 What is the Squeeze Theorem?
The squeeze theorem states that if f(x)≤g(x)≤h(x) and limx→af(x)=limx→ah(x)=L, then limx→ag(x) must also =L. Take a look at the visual below!
We can see that the function g(x) is sandwiched between f(x) and h(x), so it must follow the same rule in the shown interval. limx→ag(x)=L
🧮 Squeeze Theorem Practice Problems
Let’s work on a few questions and make sure we have the concept down!
1) Squeeze Theorem Logic
Functions g and h are twice-differentiable functions with g(2)=h(2)=4. It is known that g(x)<h(x) for 1<x<3. Let k be a function satisfying g(x)≤k(x)≤h(x) for 1<x<3. Is k continuous at x=2? Justify your answer.
Once you’re ready, keep on reading to see how to approach this question. ⬇️
If functions g and h are twice-differentiable, they must be continuous. Therefore, limx→2g(x)=4 and limx→2h(x)=4. Since g(x)≤k(x)≤h(x) and the conditions for continuity are met, the squeeze theorem for k(x) applies at x=2. So, limx→2k(x)=4 .
Since 4=g(2)<k(2)<h(2)=4, k(2) must equal 4.
We can then conclude that k(x) is continuous at x=4 because limx→2k(x)=k(2)=4 . Brush up on continuity rules with this guide here: Confirming Continuity Over an Interval.
This question is from the 2019 AP Calculus AB examination administered by College Board. All credit to College Board. Way to go! 👏
2) Computing a Limit Using Squeeze Theorem
Find the limit of the function g(x)=xcos(x1) as x 0, using the Squeeze Theorem.
In this case, we can use the fact that −1<cos(x1)<1 for all x to create a bounding function.
Multiply the inequality by x, and then consider the bounding functions f(x)=−x and h(x)=x so that −x<xcos(x1)<x.
Since f(x)≤g(x)≤h(x) , and the functions are known to be continuous, the Squeeze Theorem can be applied. Let’s check the limits of the bounding functions as they approach 0 to see if they squeeze g(x) at x=0.
limx→0f(x)=limx→a−x=0
limx→0h(x)=limx→ax=0
Because limx→0f(x)=limx→0h(x)=0, the Squeeze Theorem holds true, and…
x→0limg(x)=x→0limxcos(x1)=0
Check out the graph below to confirm our answer visually!
Graph proving limx→0xcos(x1)=0 by the Squeeze Theorem
Graph created with Desmos
You nailed it! This was a tough one. 💪
🌟 Closing
Great work! 🙌 The squeeze theorem is a key foundational idea for AP Calculus. You can anticipate encountering questions involving limits and the squeeze theorem on the exam, both in multiple-choice and as part of a free response.
If you’d like some steps to follow, here they are:
🤔 Identifying the Function: Recognize the function for which you need to determine the limit.
👀 Finding the 'Squeeze' Functions: Locate two functions that 'squeeze' the given function between them.
🏁 Ensuring Known Limits: Confirm that the limits of the 'squeezing' functions are known as x approaches the same value.
You got this! 🤩
Key Terms to Review (6)
Approaches: In calculus, "approaches" refers to the behavior of a function as the input values get closer and closer to a certain value or infinity. It describes how the output values of the function get arbitrarily close to a specific number or approach a particular limit.
Bounded: A function is bounded if its values are limited or restricted within a certain range. It means that the function does not go to infinity or negative infinity.
Function: A relationship between two sets where each input (domain) value corresponds to exactly one output (range) value.
Limit: The limit of a function is the value that the function approaches as the input approaches a certain value or infinity. It represents the behavior of the function near a specific point.
Oscillate: In calculus, "oscillate" refers to the behavior of a function when it repeatedly alternates between two values without approaching any particular number. It describes functions that exhibit periodic fluctuations around one or more equilibrium points.
Squeeze Theorem: The Squeeze Theorem states that if two functions, g(x) and h(x), both approach the same limit L as x approaches a certain value c, and another function f(x) is always between g(x) and h(x) near c (except possibly at c itself), then f(x) also approaches L as x approaches c.