5128_Ch08_434-471

446 Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals
EXPLORATION 1
Exploring L’Hôpital’s Rule Graphically
Consider the function f x sin x
.
x
1. Use l’Hôpital’s Rule to find lim x→0 f x.
2. Let y 1 sin x, y 2 x, y 3 y 1 y 2 , y 4 y 1 y 2 . Explain how graphing y 3 and
y 4 in the same viewing window provides support for l’Hôpital’s Rule in part 1.
3. Let y 5 y 3 . Graph y 3 , y 4 , and y 5 in the same viewing window. Based on what
you see in the viewing window, make a statement about what l’Hôpital’s Rule
does not say.
L’Hôpital’s Rule applies to one-sided limits as well.
[–1, 1] by [–20, 20]
Figure 8.7 The graph of
f x sin xx 2 . (Example 3)
EXAMPLE 3 Using L’Hôpital’s Rule with One-Sided Limits
Evaluate the following limits using l’Hôpital’s Rule:
(a) lim si n x
x→0
x2
(b) lim si n x
x→0
x2
Support your answer graphically.
SOLUTION
(a) Substituting x 0 leads to the indeterminate form 0/0. Apply l’Hôpital’s Rule by differentiating
numerator and denominator.
lim si n x
x→0 x2
lim co s x 1
x→0
2x
0
(b) lim si n x
x→0
x2
lim co s x 1
x→0
2x
0
Figure 8.7 supports the results. Now try Exercise 11.
When we reach a point where one of the derivatives approaches 0, as in Example 3, and
the other does not, then the limit in question is 0 (if the numerator approaches 0) or
infinity (if the denominator approaches 0).
Indeterminate Forms , • 0,
A version of l’Hôpital’s Rule also applies to quotients that lead to the indeterminate form
. If f x and gx both approach infinity as x→a, then
f x
f
x
lim lim ,
x→a g x
x→a g
x
provided the latter limit exists. The a here (and in the indeterminate form 00) may itself
be finite or infinite, and may be an endpoint of the interval I of Theorem 5.
EXAMPLE 4 Working with Indeterminate Form
Identify the indeterminate form and evaluate the limit using l’Hôpital’s Rule. Support
your answer graphically.
lim sec x
x→p2 1 tan x
continued