Difference between revisions of "Squeeze Theorem"
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==Theorem== | ==Theorem== | ||
− | Suppose f(x) is between g(x) and h(x) for all x in the neighborhood of S. If g and h approach some common limit L as x approaches S, then \lim_{x\to S}f(x)=L. | + | Suppose <math>f(x)</math> is between <math>g(x)</math> and <math>h(x)</math> for all <math>x</math> in the neighborhood of <math>S</math>. If <math>g</math> and <math>h</math> approach some common limit L as <math>x</math> approaches <math>S</math>, then <math>\lim_{x\to S}f(x)=L</math>. |
==Proof== | ==Proof== |
Revision as of 11:09, 7 May 2008
This is an AoPSWiki Word of the Week for May 4-11 |
The Squeeze Theorem (also called the Sandwich Theorem or the Squeeze Play Theorem) is a relatively simple theorem that deals with calculus, specifically limits.
Theorem
Suppose is between
and
for all
in the neighborhood of
. If
and
approach some common limit L as
approaches
, then
.
Proof
If is between
and
for all
in the neighborhood of
, then either
or
for all
in the neighborhood of
. Since the second case is basically the first case, we just need to prove the first case.
For all , we must prove that there is some
for which
.
Now since, , there must exist
such that,
and,
Now let . If
, then
So . Now by the definition of a limit, we get
.