# Squeeze Theorem

The **Squeeze Theorem** (also called the **Sandwich Theorem** or the **Squeeze Play Theorem**) is a relatively simple theorem that deals with calculus, specifically limits.

## Theorem

Suppose is between and for all in a neighborhood of the point . If and approach some common limit as approaches , then .

### Proof

If is between and for all in the neighborhood of , then either or for all in this neighborhood. The two cases are the same up to renaming our functions, so assume without loss of generality that .

We must show that for all there is some for which implies .

Now since , there must exist such that

Now let . If then

So . Now by the definition of a limit we get as desired.

## Applications and examples

The Squeeze Theorem can be used to evaluate limits that might not normally be defined. An example is the limit $\lim_{x\to\0} f(x)=x^2e^{\sin\frac{1}{x}}$ (Error compiling LaTeX. ! Undefined control sequence.). The limit is not normally defined, because the function oscillates infinitely many times around 0, but it can be evaluated with the Squeeze Theorem as following. Create two functions, and . It is easy to see that around 0, the function in question is squeezed between these two functions, and the limit as both of these approach 0 is 0, so $\lim_{x\to\0} x^2e^{\sin\frac{1}{x}}$ (Error compiling LaTeX. ! Undefined control sequence.) is 0.