5128_Ch08_434-471

f ( x)
7. lim lim
x→∞ g ( x)
x→∞ 1 ln x
x
1 0 1
Quick Review 8.3 (For help, go to Sections 2.2 and 4.1.)
Section 8.3 Relative Rates of Growth 457
9. (a) Local minimum at (0, 1) Local maximum at (2, 1.541)
In Exercises 1–4, evaluate the limit.
In Exercises 7 and 8, show that g is a right end behavior model for f.
x
7. gx x, f x x ln x
8. gx 2x, f x 4x 2 5x lim f ( x)
lim
2
x→ e
x2 x 0
9. Let f x e x x
. 2
x→∞ g(
x)
x→∞ 1 5
4x
1
e
x
Find the
(a) local extreme values of f and where they occur.
(b) intervals on which f is increasing. [0, 2]
(c) intervals on which f is decreasing. (, 0] and [2, )
10. Let f x x sin x
.
3x 1
x
2x 2
1. lim ln x 0 2. lim
x→ e x
x→ x
e3
3. lim x2
x→ e2x 4. lim
In Exercises 5 and 6, find an end behavior model (Section 2.2) for
the function.
5. f x 3x 4 5x 3 x 1 3x 4
6. f x 2x3
x 2
13. lim lo g x
1
x→∞ ln
x 2ln 10
Section 8.3 Exercises
Find the absolute maximum value of f and where it occurs.
f doesn’t have an absolute maximum value. The values are always less than 2 and the
values get arbitrarily close to 2 near x 0, but the function is undefined at x 0.
same
In Exercises 1–4, show that e x grows faster than the given function. In Exercises 31–34, show that the three functions grow at the
1. x 3 3x 1 2. x 20 x
lim ex
lim x→ x
e2 0
rate as x→.
x→∞ x 3 ∞
3x 1
3. e cos x ex
lim
x→ e
co s x
4. 52 x ex
lim x→ (5 2)
31. f 1 x x, f 2 x 10x 1, f 3 x x 1
30. 2 x , x 2 , ln 2 x , e x (ln 2) x , x 2 ,2 x , e x (Hint: What is the nth derivative of x n ?)
In Exercises 5–8, show that ln x grows slower than the given function. 32. f 1 x x 2 , f 2 x x 4 x, f 3 x x 4 x
3
5. x ln x lim
ln x
0
x→ x ln x
6. x lim l n x
0
x→ 33. f
x
1 x 3 x , f 2 x 9 x 2 x , f 3 x 9 x 4 x
7. 3 x lim l n x
0 8. x 3
x→
3
lim ln x 0
x
x→ x 3 34. f 1 x x 3 , f 2 x x 4 2x 1
, f
In Exercises 9–12, show that x 2 grows at the same rate as the given
x 1
3 x 2 x5
1
x2
1
function.
9. x 2 4x
lim
4x
x→ x2 1
x 2 10. x x
4 5 lim In Exercises 35–38, only one of the following is true.
x x 4 5
x→ x2
1
i. f grows faster than g.
11. 3 x 6 x
lim x 6 x
2 1 12. x 2 sin x lim sin x
x→ x 2 x→ x 2 1
ii. g grows faster than f.
In Exercises 13 and 14, show that the two functions grow at the same iii. f and g grow at the same rate.
rate.
Use the given graph of fg to determine which one is true.
13. ln x, logx 14. e x1 , e x lim ex 1
e
x→ ex
35. 36.
In Exercises 15–20, determine whether the function grows faster than
e x , at the same rate as e x , or slower than e x as x→.
15. 1 x 4 Slower 16. 4 x Faster
17. x ln x x Slower 18. xe x Faster
19. x 1000 Slower 20. e x e x 2 Same rate
[0, 100] by [–1000, 10000]
[0, 10] by [–0.5, 1]
f grows faster than g
g grows faster than f
In Exercises 21–24, determine whether the function grows faster than
x 2 , at the same rate as x 2 , or slower than x 2 as x→.
37. 38.
21. x 3 3 Faster 22. 15x 3 Slower
23. ln x Slower 24. 2 x Faster
In Exercises 25–28, determine whether the function grows faster than
ln x, at the same rate as ln x, or slower than ln x as x→.
[0, 100] by [–1, 1.5]
[0, 20] by [–1, 3]
25. log 2 x 2 Same rate 26. 1x Slower
f and g grow at the same rate f and g grow at the same rate
27. e x Slower 28. 5 ln x Same rate
Group Activity In Exercises 39–41, do the following
comparisons.
In Exercises 29 and 30, order the functions from slowest-growing to 39. Comparing Exponential and Power Functions
fastest-growing as x→.
(a) Writing to Learn Explain why e x grows faster than
29. e x , x x , ln x x , e x2 e x2 , e x , (ln x) x , x x
x n as x→ for any positive integer n, even n 1,000,000.