##
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, *x*^{2} and -*x*^{2} 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.

**Answer.**

**Exercise 2.** Use the Sandwich Theorem to prove that

**Answer.**

**Exercise 3.** Consider the function

Use the Sandwich Theorem to prove that

**Answer.**

**
**

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