# 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 function with the limit . 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 is 0.

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