Difference between revisions of "Squeeze Theorem"
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== Applications and examples== | == 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 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. |
Revision as of 21:16, 28 August 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
![\[|x-S|<\delta_1 \Rightarrow |g(x)-L|<\varepsilon \textrm{ and } |x-S|<\delta_2 \Rightarrow |h(x)-L|<\varepsilon.\]](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
The Squeeze Theorem can be used to evaluate limits that might not normally be defined. An example is the function 
with the limit $\lim_{x\to\0} f(x)$ (Error compiling LaTeX. Unknown error_msg). 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 $\lim_{x\to\0} f(x)$ (Error compiling LaTeX. Unknown error_msg) is 0.
