Difference between revisions of "Squeeze Theorem"
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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. | 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. | ||
Revision as of 20:50, 19 May 2008
The Squeeze Theorem (also called the Sandwich Theorem or the Squeeze Play Theorem) is a relatively simple theorem that deals with calculus, specifically limits.
Contents
[hide]Theorem
Suppose is between
and
for all
in a neighborhood of the point
. If
and
approach some common limit
as
approaches
, then
.
Proof
If is between
and
for all
in the neighborhood of
, then either
or
for all
in this neighborhood. The two cases are the same up to renaming our functions, so assume without loss of generality that
.
We must show that for all there is some
for which
implies
.
Now since , there must exist
such that
Now let . If
then
So . Now by the definition of a limit we get
as desired.
Applications and examples