Squeeze Theorem
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
The Squeeze Theorem can be used to evaluate limits that might not normally be defined. An example is the function with thelimit $\lim_{x\to\0} f(x)=x^2 e^{\sin\frac{1}{x}}$ (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 $\lim_{x\to\0} x^2 e^{\sin\frac{1}{x}}$ (Error compiling LaTeX. Unknown error_msg) is 0.