## The Pinching or Sandwich Theorem

As a motivation let us consider the function

When x get closer to 0, the function fails to have a limit. So we are not able to use the basic properties discussed in the previous pages. But we know that this function is bounded below by -1 and above by 1, i.e.

for any real number x. Since , we get

Hence when x get closer to 0, x2 and -x2 become very small in magnitude. Therefore any number in between will also be very small in magnitude. In other words, we have

This is an example for the following general result:

Theorem: The "Pinching" or "Sandwich" Theorem Assume that

for any x in an interval around the point a. If

then

Example. Let f(x) be a function such that , for any . The Sandwich Theorem implies

Indeed, we have

which implies

for any . Since

then the Sandwich Theorem implies

Exercise 1. Use the Sandwich Theorem to prove that

for any a > 0.

Use the Sandwich Theorem to prove that

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Mohamed A. Khamsi
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