Understanding the Squeeze Theorem in Calc II

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In summary, the squeeze theorem is used to evaluate limits by identifying a piece of the function that can be bounded and then using the bounds to create an inequality that can be solved using the squeeze theorem. This is demonstrated in the example of \frac{x^2 + \ln x}{x^2 + 1}, where the bounds 0 < \ln x < x are used to create the inequality \frac{x^2}{x^2+1} < \frac{x^2 + \ln x}{x^2 + 1} < \frac{x^2 + x}{x^2+1} and the limits on both sides are evaluated to be 1. This shows that the original limit is also 1
  • #1
kdinser
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Ok, this is kind of embarrassing. I'm halfway through calc II and while the squeeze theorem makes sense to me when I read it, I don't see how it's applied. I feel like I must be missing something very fundamental for this to not make sense.

[tex]\lim_{x \rightarrow \infty}(cosx)/x = 0 [/tex]


What are the 2 functions that are squeezing the 3rd? Once you know the 2 functions do just evaluate one of them at the limit?

Is it as simple as just splitting up the function? [tex](1/x)(cos x)[/tex] and then evaluating [tex]1/x[/tex] at infinity. That would certainly yield 0.
 
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  • #2
Usually, you use the squeeze theorem by identifing a piece of your function that you can bound.

Let me do an example in a slightly silly way:

[tex]
\lim_{x \rightarrow \infty} \frac{x^2 + \ln x}{x^2 + 1}
[/tex]

Now, when x is large, we have the bound [itex]0 < \ln x < x[/itex]. With a little work, we can prove:

[tex]
\frac{x^2}{x^2+1} < \frac{x^2 + \ln x}{x^2 + 1} < \frac{x^2 + x}{x^2+1}
[/tex]

for large x. (such as x > 1)

We can evaluate the limits on the left and right hand sides: they both come out to 1. So, the original limit must also be 1!


That's how the squeeze theorem is usually used: you identify a piece of the function that you can bound, then substitute in the bounds to get an inequality to which you can apply the squeeze theorem.
 
  • #3


First of all, there is no need to feel embarrassed. Calculus is a challenging subject and it's completely normal to have trouble understanding certain concepts. The squeeze theorem can be a bit tricky to grasp at first, but with some practice and examples, you will soon have a better understanding of it.

To answer your question, the two functions that are "squeezing" the third one in this particular example are f(x) = cosx and g(x) = -1/x. These two functions are essentially bounding the third function, h(x) = (cosx)/x, from above and below. As x approaches infinity, f(x) and g(x) both approach 0, which means that h(x) must also approach 0.

You are correct in thinking that evaluating one of the functions at the limit is a key step in using the squeeze theorem. In this case, we can evaluate g(x) = -1/x at x = infinity, which gives us a limit of 0. Since h(x) is always between f(x) and g(x), it must also approach 0 as x approaches infinity.

So, in essence, yes, it is as simple as splitting up the function and evaluating one of the functions at the limit. However, it's important to remember that the other function must also approach the same limit in order for the squeeze theorem to be applicable.

I hope this helps clear up any confusion you may have had about the squeeze theorem. Keep practicing and seeking help when needed, and you will soon have a strong understanding of calculus concepts like this one. Good luck!
 

1. What is the Squeeze Theorem in Calculus II?

The Squeeze Theorem, also known as the Sandwich Theorem or the Pinching Theorem, is a fundamental theorem in Calculus II that helps us determine the limit of a function by using the limits of two other functions that "squeeze" the original function between them. It is commonly used to evaluate limits of complicated functions that are difficult to solve directly.

2. How does the Squeeze Theorem work?

The Squeeze Theorem states that if we have three functions, f(x), g(x), and h(x), such that f(x) ≤ g(x) ≤ h(x) for all x in some interval around a point c, and the limits of f(x) and h(x) as x approaches c are equal to some value L, then the limit of g(x) as x approaches c is also equal to L. In other words, if we have a "sandwich" of functions, and the top and bottom layers are approaching the same limit, then the middle layer must also approach that limit.

3. Why is the Squeeze Theorem important?

The Squeeze Theorem is an important tool in Calculus II because it allows us to evaluate limits of complicated functions that cannot be solved using direct substitution or algebraic manipulation. It also helps us understand the behavior of functions near a particular point and determine if a function is approaching a finite or infinite limit.

4. Can the Squeeze Theorem be used to evaluate limits at infinity?

Yes, the Squeeze Theorem can be used to evaluate limits at infinity. In this case, we would use a similar approach, but instead of looking at the limit of a function as x approaches a specific point, we would look at the limit as x approaches either positive or negative infinity. The same concept applies - if we can "squeeze" the function between two other functions that have limits at infinity, then the middle function must also have the same limit at infinity.

5. Are there any common mistakes when using the Squeeze Theorem?

One common mistake when using the Squeeze Theorem is forgetting to check the limits of the top and bottom functions. It is important to make sure that the limits of these functions exist and are equal before using the Squeeze Theorem. Additionally, it is important to make sure that the middle function is actually sandwiched between the top and bottom functions - if there is a gap between the functions, the Squeeze Theorem cannot be used.

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