Introduction
We want to prove that as
x
approaches
a
,
f(x)
approaches
c
.
We have learned one tool for proving this, the Epsilon-Delta Method. The second tool in our arsenal is the Squeeze Theorem.
The squeeze theorem states that if we can sandwich a function between two other functions, and the two outer functions approach the same limit, then the middle function must also approach that limit.
In mathematical notation, if
\lim_{x \to a} g(x) = c, \lim_{x \to a} h(x) = c
, and
g(x) \leq f(x) \leq h(x)
, then
\lim_{x \to a} f(x) = c
.
Below is an example.
f(x)=x^2 \sin(\frac{1}{x})
.
g(x)=-x^2
.
h(x)=x^2
.
g(x) \leq f(x) \leq h(x)
.
\lim_{x \to 0} g(x) = \lim_{x \to 0} h(x) = 0
.
Therefore, \lim_{x \to 0} f(x) = 0
.
For the squeeze theorem, we do not need
g(x) \leq f(x) \leq h(x)
for all values of x. We only need it to be able to define an interval around
a
where it is true.
For example,
\lim_{x \to 0} \frac{\sin(x)}{x} = 1
.
f(x)=\frac{\sin(x)}{x}
.
g(x)=cos(x)
.
h(x)=1
. When
x=5
,
g(x) \gt f(x)
.
However, g(x) \leq f(x) \leq h(x), \forall x \in (-1,1)
\lim_{x \to 0} g(x) = \lim_{x \to 0} h(x) = 1
.
Therefore, \lim_{x \to 0} f(x) = 1
.