Section 1.8 The Squeeze Theorem
Objectives
- Evaluate the limit of a function by using the squeeze theorem.
- Articulate the difference between a one-sided and two-sided limit.
- Evaluate two-sided limits using two one-sided limits.
limx→2(x−2)(x−3)x−2
we recognize that
(x−2)(x−3)x−2=x−3
at all values of x except possibly at x=2. So the two functions have the same limit there, and we can use limit laws to evaluate
limx→2x−3=2−3=−1.
The squeeze theorem combines this idea with approximation. Suppose we have a limit we want to find,
limx→0xcos(x).
We want to find two functions for which we can compute the limit, call them f(x) and h(x) that bracket our function. In other words we want
f(x)≤xcos(x)≤h(x)
for all values nearby to zero. If limx→0f(x)=L and limx→0h(x)=M and we take the limit of all the parts of this set of inequalities, then we get
L=limx→0f(x)≤limx→0xcos(x)≤limx→0h(x)=M,
or
L≤limx→0xcos(x)≤M.
Now this tells us something about the limit - it is between L and M - but still doesn't give us the complete answer. The trick now is to find an f(x) and g(x) so that L=M - then the computation above would imply that limx→0xcos(x)=L.
But how in the world can we find these functions? Sadly, there is no formula or recipe for doing this - we need to think about x and cos(x) and see what we can figure out. Let's start by looking at the graph of cos(x) over one of its periods in Figure 1.8.1. 
−1⋅x≤xcos(x)≤1⋅x,
and for negative values of x, we have
1⋅x≤xcos(x)≤−1⋅x.
This makes this a little complicated - to help us we will introduce a new notion, the one sided limit. When we made our somewhat informal definition of the limit, we thought about what value the function tended to as we approached the x value from both sides. A one-sided limit has a similar definition but we only consider values from the right or left of the limiting x value. We use a modified notation as well:
limx→a+f(x)
when we only look at values to the right of a and
limx→a−f(x)
when we only consider values to the left. The two-sided limit exists if both one-sided limits exist and are equal to the same value.
Let's apply this idea here to try to find limx→0xcos(x). First, we'll look at values to the right of zero - positive values of x. Taking the one-sided limit of the series of inequalities, we have
limx→0+−x≤limx→0+xcos(x)≤limx→0+x.
Now both the functions f(x)=−x and h(x)=x are polynomials so we can find their limits by evaluation:
limx→0+−x=0=limx→0+x.
Since these are equal, we conclude
limx→0+xcos(x)=0.
If we only consider values to the left- negative x values- we have a similar computation:
0=limx→0−−x≤limx→0−xcos(x)≤limx→0−x=0,
and we conclude that
limx→0−xcos(x)=0
as well. Taken together, this tells us that the limit of xcos(x) as x approaches zero is zero.
This is the essence of the Squeeze Theorem: Theorem 1.8.2. The Squeeze Theorem.
Let f(x),g(x), and h(x) be defined for all x≠a over an open interval containing a. If
f(x)≤g(x)≤h(x)
for all x≠a in an open interval containing a and
limx→af(x)=L=limx→ah(x),
where L is a real number, then limx→af(x)=L.