5128_Ch08_434-471

450 Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals
Applying l’Hôpital’s Rule to ln f x we obtain
lim ln f x lim ln x
x→ x→ x
lim 1 x
x→ 1
Differentiate.
Therefore,
lim
x→
1 x 0.
lim
x→ x1x lim f x lim e ln f x e 0 1.
x→ x→
Now try Exercise 25.
Quick Review 8.2 (For help, go to Sections 2.1 and 2.2.)
In Exercises 1 and 2, use tables to estimate the value of the limit.
1. lim
x→ ( 1 0 .1 1.1052 2. lim
x )x
x1ln x 2.7183
x→0
In Exercises 3–8, use graphs or tables to estimate the value of the
limit.
3. lim
x→0 (
x
1 1 x )
x
1 4. lim
x→1 ( 1 1 x )
t 1
5. lim 2 6. lim 4 x 2 1
2
t→1 t 1
x→ x 1
7. lim sin 3x
3 8. lim
x→0 x
tan u
1
u→p2 2 tan u
In Exercises 9 and 10, substitute x 1h to express y as a function
of h.
9. y x sin 1 x
x y sin h
y (1 h) 1h
h
10. y ( 1 1 x )
Section 8.2 Exercises
In Exercises 1–4, estimate the limit graphically and then use l’Hôpital’s
Rule to find the limit.
2
1. lim
x→2 x
x2
2. lim sin (5x)
4
x→0 x
3. lim 2 x 2
4. lim 3 x 1
x→2 x 2
x→1 x 1
In Exercises 5–8, apply the stronger form of l’Hôpital’s Rule to find
the limit.
5. lim 1 cos x
1 sin
12
6. lim 14
x→0 x2
u→p/2 1 cos
(2)
cos t 1
x
7. lim t→0 e
t
–1
8. lim 2 4x
4
t 1
x→2 x
3
16
12x
16
In Exercises 9–12, use l’Hôpital’s Rule to evaluate the one-sided
limits. Support your answer graphically.
9. (a) lim s in
4x
2 (b) lim
x→0
sin
2x
s in
4x
2
x→0
sin
2x
10. (a) lim tan x
1 (b) lim
x→0
x
tan x
1
x→0
x
11. (a) lim si n x
(b) lim
x→0
x3
si n x
x→0
x3
12. (a) lim ta n x
(b) lim
x→0
x2
ta n x
x→0
x2
In Exercises 13–16, identify the indeterminate form and evaluate the
limit using l’Hôpital’s Rule. Support your answer graphically.
csc x Left ()(),
13. lim
14. lim
x→p 1 cot x
1 sec x Left ()(),
right ()(),
x→p2 tan x
right ()(),
limit 1
limit 1
15. lim ln ( x 1)
()(), limit ln 2
x→ log2 x
16. lim 5 x2
3x
x→ 7x2
()(), limit 57
1
In Exercises 17–26, identify the indeterminate form and evaluate the
limit using l’Hôpital’s Rule.
17. lim (x ln x) • 0, 0 18. lim
x→0 x→ x tan 1 x • 0, 1
19. lim csc x cot x cos x 20. lim ln (2x) ln x 1
x→0 x→
,1
21. lim e x x 1x
22. lim x 1x1 , ln 2
x→0 1 , e 2 x→1 1 , e
23. lim
x→1
x 2 2x 1 x1 0 0 ,1 24. lim
x→0 sin x x 0 0 ,1
25. lim
x→0 (
x
1 1 x )
0 ,1
26. lim
x→
ln x 1x 0 ,1
In Exercises 27 and 28, (a) complete the table and estimate the limit.
(b) Use l’Hôpital’s Rule to confirm your estimate.
x 5
27. lim f x, f x ln
x→ x
x | 10 | 10 2 | 10 3 | 10 4 | 10 5
f x | | | | |
28. lim f x, f x x sin x
x→0
x3
x | 10 0 | 10 1 | 10 2 | 10 3 | 10 4
f x | | | | |