# 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== | ||

+ | {{incomplete|section}} | ||

− | |||

==See Also== | ==See Also== |

## Revision as of 21:50, 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.

## 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