Expert: Ahmed Salami - 4/20/2009

Question
What is the purpose of squeeze theorem and its application in the real world?

Answer
Hi Lubabalo,
The Squeeze Theorem is a theorem regarding the limit of a function. It is used in an effort to simplify the computation of otherwise complicated limits.
Suppose we have a function f(x) whose limit is difficult to find but we know that its value is always inclusively in between the values of two other functions g(x) and h(x) i.e
g(x) <= f(x) <= h(x)
and lets say that we know that the limits of g(x) and h(x) at a particular value x = a to be equal to some L, then the limit of f(x) at x = a is also L. This is the Squeeze Theorem.
As an example, lets say that we want to find the limit of sinx/x as x approaches 0. We know that for
0 < x < #/2       where # represents pi
sinx < x < tanx
considering sinx < x and dividing both sides by x
sinx/x < 1
considering x < tanx and multiplying both sides by cosx
x.cosx < sinx
sinx/x > cosx
combining the two inequalities gives
cosx < sinx/x < 1
We can also show this for when -#/2 < x < 0 and therefore show that for -#/2 < x < #/2,
cosx < sinx/x < 1
But we know that as x approaches 0, cosx approaches 1 and so definitely the limit of sinx/x as x approaches 0 is 1. This is an application of the Squeeze Theorem.

Hope it helps you.

Regards