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 | ||
[[Image:Squeeze theorem example.jpg|thumb|Squeeze Theorem]] | [[Image:Squeeze theorem example.jpg|thumb|Squeeze Theorem]] | ||
==Theorem== | ==Theorem== | ||
− | Suppose <math>f(x)</math> is between <math>g(x)</math> and <math>h(x)</math> for all <math>x</math> in | + | Suppose <math>f(x)</math> is between <math>g(x)</math> and <math>h(x)</math> for all <math>x</math> in a [[neighborhood]] of the point <math>S</math>. If <math>g</math> and <math>h</math> approach some common limit <math>L</math> as <math>x</math> approaches <math>S</math>, then <math>\lim_{x\to S}f(x)=L</math>. |
− | ==Proof== | + | ===Proof=== |
− | If <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>, then either <math>g(x) | + | If <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>, then either <math>g(x)\leq f(x) \leq h(x)</math> or <math>h(x)\leq f(x)\leq g(x)</math> for all <math>x</math> in this neighborhood. The two cases are the same up to renaming our [[function]]s, so assume without loss of generality that <math>g(x)\leq f(x) \leq h(x)</math>. |
− | + | We must show that for all <math>\varepsilon >0</math> there is some <math>\delta > 0</math> for which <math>|x-S|<\delta</math> implies <math>|f(x)-L|<\varepsilon</math>. | |
+ | |||
+ | Now since <math>\lim_{x\to S}g(x)=\lim_{x\to S}h(x)=L</math>, there must exist <math>\delta_1,\delta_2>0</math> such that | ||
+ | |||
+ | <cmath>|x-S|<\delta_1 \Rightarrow |g(x)-L|<\varepsilon \textrm{ and } |x-S|<\delta_2 \Rightarrow |h(x)-L|<\varepsilon.</cmath> | ||
+ | |||
+ | Now let <math>\delta = \min\{\delta_1,\delta_2\}</math>. If <math>|x-S|<\delta</math> then | ||
+ | |||
+ | <math>-\varepsilon < g(x) - L \leq f(x) - L \leq h(x) - L < \varepsilon.</math> | ||
+ | |||
+ | So <math>|f(x)-L|<\varepsilon</math>. Now by the definition of a limit we get <math>\lim_{x\to S}f(x)=L</math> as desired. | ||
+ | |||
+ | == Applications and examples== | ||
+ | The Squeeze Theorem can be used to evaluate limits that might not normally be defined. An example is the function <math>f(x)=x^2 e^{\sin\frac{1}{x}}</math> with the limit <math>\lim_{x\to 0} f(x)</math>. The limit is not normally defined, because the function oscillates infinitely many times around 0, but it can be evaluated with the Squeeze Theorem as following. Create two functions, <math>x^2</math> and <math>-x^2</math>. It is easy to see that around 0, the function in question is squeezed between these two functions, and the limit as both of these approach 0 is 0, so <math>\lim_{x\to 0} f(x)</math> is 0. | ||
+ | |||
+ | {{stub}} | ||
==See Also== | ==See Also== |
Latest revision as of 15:36, 1 December 2015
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 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
The Squeeze Theorem can be used to evaluate limits that might not normally be defined. An example is the function with the limit
. The limit is not normally defined, because the function oscillates infinitely many times around 0, but it can be evaluated with the Squeeze Theorem as following. Create two functions,
and
. It is easy to see that around 0, the function in question is squeezed between these two functions, and the limit as both of these approach 0 is 0, so
is 0.
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