Difference between revisions of "Squeeze Theorem"
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{{WotWAnnounce|week=May 4-11}} | {{WotWAnnounce|week=May 4-11}} | ||
− | The '''Squeeze | + | The '''Squeeze Theorem''' (also called the '''Sandwich Theorem''' or the '''Squeeze Play Theorem''') is a relatively simple [[theorem]] that deals with [[calculus]], specifically [[limit]]s. |
[[Image:Squeeze theorem example.jpg|thumb|Squeeze Theorem]] | [[Image:Squeeze theorem example.jpg|thumb|Squeeze Theorem]] |
Revision as of 19:07, 4 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.
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,
.