Difference between revisions of "Squeeze Theorem"
| Line 5: | Line 5: | ||
==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)\leq f(x) \leq h(x)</math> or <math>h(x)\leq f(x)\leq g(x)</math> for all <math>x</math> in the | + | 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 | + | 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> |
| − | <math> | + | 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> | |
| − | <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 11:14, 7 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 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
![\[|x-S|<\delta_1 \Rightarrow |g(x)-L|<\varepsilon \textrm{ and } |x-S|<\delta_2 \Rightarrow |h(x)-L|<\varepsilon.\]](http://latex.artofproblemsolving.com/9/c/0/9c0b9eee7b3368e1fef612e618fbe1fc05954332.png)
Now let 
. If then
So . Now by the definition of a limit we get as desired.
Applications and examples
