Find limits using the Squeeze Theorem.
Squeeze Theorem
In this section we find limits using the Squeeze Theorem.
The Squeeze Theorem Suppose that the compound inequality holds for all values of
in some open interval about , except possibly for itself. If then we can conclude that
as well.
example 1 Suppose for all except . Find
Since and we can use the Squeeze Theorem to conclude that as well.
(problem 1) Suppose that for all except . Find First, we find the limits of the
bounds: Since these answers are the same, the Squeeze Theorem allows us to
conclude that
example 2 Find Since is undefined, plugging in does not give a definitive answer.
Using the fact that for all values of , we can create a compound inequality for the
function and find the limit using the Squeeze Theorem. To begin, note
that for all values of except . Multiplying this compound inequality by the
non-negative quantity, , we have for all values of except . Next, note that and
Finally, by the Squeeze Theorem, we can conclude that as well. The graph
below also shows that the limit is zero. Zoom in on the origin to get the full
effect.
(problem 2a) Find using the Squeeze Theorem.
First, we need to find bounds. Since for all , for all except . Next, we need to find the limits of those bounds: Since these answers are the same, the Squeeze Theorem allows us to conclude that
First, we need to find bounds. Since for all , for all except . Next, we need to find the limits of those bounds: Since these answers are the same, the Squeeze Theorem allows us to conclude that
The Squeeze Theorem can also be used if .
(problem 2b) Find using the Squeeze Theorem.
First, we need to find bounds. Since for all , for all . Next, we need to find the limits of those bounds: Since these answers are the same, the Squeeze Theorem allows us to conclude that
First, we need to find bounds. Since for all , for all . Next, we need to find the limits of those bounds: Since these answers are the same, the Squeeze Theorem allows us to conclude that
example 3 Use the Squeeze Theorem to find a special limit: Consider the figure
below. It consists of a small triangle, a sector of a circle of radius, , and a large
triangle.
The area of the small triangle is The area of the sector is The area of the large triangle is
We can use the areas of these figures to create a compound inequality like the one found in the Squeeze Theorem. Since the area of the small triangle is less than the area of the sector which is less than the area of the large triangle, we have:
Multiply through by 2:
Divide through by . Note that if is a small positive angle, then so the direction of the inequality symbols remains unchanged:
Next we take reciprocals (this will change the direction of the inequality symbols):
which is equivalent to
We now compute the limits of the upper and lower bounds:
Since the above limits are equal, by the Squeeze Theorem
To compute the left-hand limit, we recall that for any angle . Therefore, which implies that the left-hand limit and the right-hand limit are equal:
Thus the two-sided limit exists and