# How Can I Apply the Squeeze Theorem Without Trig Functions?

• TitoSmooth
In summary: But what I'm still wondering is what is the significance of the two functions?In summary, there are two functions that bound 1+x2, but what is the significance of them?
TitoSmooth
Not sure how to apply the Squeeze Theorem when not given in trig functions.

My question is.

Lim (x^2+1)=1
x→0

not sure what values let's call them K. ie -K≤x^2+≤K.

for instances when I have.

Lim xsin(1/x)=0
x→0

i say. -1≤sin(1/x)≤1

then multiply the whole inequality by x.

-x≤xsin(1/x)≤x

therefore limit as x approaches 0 of sin(1/x)=0

how would I do it for for non trig functions?

Hi TitoSmooth!

(try using the X2 button just above the Reply box )

You probably wouldn't need it for non-trig (or non-algebraic) functions!

For example, there's no way of applying it to x2+1.

wikipedia has an example, involving two variables:

-|y| ≤ x2y/(x2 + y2) ≤ |y|​

tiny-tim said:
For example, there's no way of applying it to x2+1.
Yes, there is. There are two very simple functions which bound 1+x2 from above and from below on the interval [-1,1]. The key is that you only need to concern yourself with an interval containing x=0. You don't have to find a simple function that bounds 1+x2 from above for all x.

vela said:
Yes, there is. There are two very simple functions which bound 1+x2 from above and from below on the interval [-1,1]. The key is that you only need to concern yourself with an interval containing x=0. You don't have to find a simple function that bounds 1+x2 from above for all x.

Layman terms my man. So I could understand better. Thanks

Layman terms? I can see no technical terms in what Vela wrote, except possibly "interval" and vela defined it: "the interval [-1, 1]".

HallsofIvy said:
Layman terms? I can see no technical terms in what Vela wrote, except possibly "interval" and vela defined it: "the interval [-1, 1]".

You don't have to find a simple function that bounds 1+x2 from above for all x.

missread this. I understand now between the closed interval of -1 and 1

## 1. What is the Squeeze Theorem Problem?

The Squeeze Theorem Problem, also known as the Squeeze Theorem or the Sandwich Theorem, is a mathematical theorem used to evaluate the limit of a function by comparing it to two other functions whose limits are known. It is commonly used in calculus to prove the convergence of a sequence or series.

## 2. How does the Squeeze Theorem work?

The Squeeze Theorem states that if f(x) ≤ g(x) ≤ h(x) for all x in a certain interval, and if the limits of f(x) and h(x) are equal as x approaches a certain value, then the limit of g(x) must also be equal to that value. In other words, if two functions "squeeze" a third function, the limit of the squeezed function must also be the same as the limits of the squeezing functions.

## 3. What is the significance of the Squeeze Theorem?

The Squeeze Theorem is important because it allows us to evaluate the limit of a function even when direct substitution is not possible. It also provides a way to prove the convergence of a sequence or series, which is useful in many mathematical applications.

## 4. How do you use the Squeeze Theorem to solve problems?

To use the Squeeze Theorem, you must first identify a function whose limit you want to find. Then, find two other functions that "squeeze" the original function and whose limits are known. Substitute the values of the two functions into the Squeeze Theorem formula to find the limit of the original function.

## 5. Can the Squeeze Theorem be used for any type of function?

No, the Squeeze Theorem can only be used for functions that have a "squeeze" between two other functions. It does not work for all types of functions and may not be applicable in certain situations. It is important to carefully consider the functions being used and ensure that they satisfy the requirements of the Squeeze Theorem before using it to solve problems.

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