Difference between revisions of "Squeeze Theorem"
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The '''Squeeze Play Theorem''' (also called the '''Squeeze Theorem''' or the '''Sandwich Theorem''') is a relatively simple [[theorem]] that deals with [[calculus]], specifically [[limit]]s. | The '''Squeeze Play Theorem''' (also called the '''Squeeze Theorem''' or the '''Sandwich Theorem''') is a relatively simple [[theorem]] that deals with [[calculus]], specifically [[limit]]s. | ||
Revision as of 21:08, 3 May 2008
This is an AoPSWiki Word of the Week for May 4-11 |
The Squeeze Play Theorem (also called the Squeeze Theorem or the Sandwich 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.
If increases to
, then
goes to either
or
, where
. If
decreases to
, then
goes to either
or
, where
. Since
can't go to
or
, then
must go to
. Therefore,
.