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