## 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 [Back] [Next]
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Mohamed A. Khamsi
Helmut Knaust