The Squeeze Theorem

Suggested Prerequesites: Formal look at limits

Our main motivation for the squeeze theorem is to so that we can evaluate the following limits, which are necessary in determining the derivatives of sin and cosine.

(If you can read this, then it means that either your browser is incapable of viewing Java applets or you have turned off this capability. This may make reading these pages harder. I have tried to compensate for this as much as possible) The limit as x approaches zero of sin x is zero. and The limit as x approaches zero of cos x is 1.

The squeeze theorem is applied to these very useful limits on the page Useful Trig Limits.
The Squeeze Theorem:

If there exists a positive number p with the property that

g(x) is less than or equal to f(x) which is less than or equal to h(x)

for all x that satisfy the inequalities 0 < |x-a| < p , and if The limit as x approaches a of g(x) is L and The limit as x approaches a of h(x)> is L , then

The limit as x approaches a of f(x) is L
Proof:

This statement is sometimes called the "squeeze theorem" because it says that a function squeezed between two functions approacing the same limit L must also approach L.

Intuitively, this means that the function f(x) gets squeezed between the other functions. Since g(x) and h(x) are equal at x = a, it must also be the case that f(x) = g(x) = h(x) = L at x = a, since there is no room for x to be anything else.

For the formal proof, let e > 0 be given, and chose positive numbers d1 and d2 so that

0 < |x-a| < d1 implies L-e < g(x) < L+e
and
0 < |x-a| < d2 implies L-e < h(x)

Define to be the smallest of the numbers . Then implies

so and the proof is complete.

The Squeeze Theorem applied to Trig Limits | Back to the Calculus page | Back to the World Web Math top page
jjnichol@mit.edu
Last Modified 23 June 1997