Thomas Calculus 13th [Solutions]

Section 7.4 Relative Rates of Growth 533
( )
13. When the degree of f is less than or equal to the degree of g since lim f x
g( x)
0 when the degree of f is smaller
x
( )
than the degree of g, and lim f x a
(the ratio of the leading coefficients) when the degrees are the same.
x
g( x)
b
14. Polynomials of a greater degree grow at a greater rate than polynomials of a lesser degree. Polynomials of the
same degree grow at the same rate.
15.
1
x 1
1
x
ln( x 1) x 1
x
ln x
x x
x 1
x
1
lim lim lim lim 1 and
1
x 999
1
x
ln( x 999)
x
x
ln x
x x
x 999
lim lim lim 1
16.
1
x a
1
x
ln( x a) x 1
x
ln x
x x
x a
x
1
lim lim lim lim 1. Therefore, the relative rates are the same.
17.
10x
1
lim lim
10x
1
x 1
10 and lim lim
x 1
1 1.
x x x
x
x x x
x
conclude that 10x 1 and x 1 have the same growth rate (that of x ).
Since the growth rate is transitive, we
18.
4 4
2 4
x x x x
x x x
lim lim 1 and
x
conclude that
4 3 4 3
2 4
x x x x
x x x
lim lim 1. Since the growth rate is transitive, we
x
4 4 3
2
x x and x x have the same growth rate (that of x ).
19.
n
n 1
x nx n!
x
x
e x
x
e x
x
e
lim lim lim 0
n
x
x o e for any non-negative integer n
n
n 1
20. If p( x) an
x an
1x a1x a 0,
then
p( x) 1
lim a n
n
lim
x a lim
x a lim
x a lim
1
where each limit is zero (from Exercise
21. (a)
n n 1 ex 1 0
x e x e x x e x e
p( x)
x
lim 0
x
x e
x x x x
19). Therefore,
e grows faster than any polynomial.
1/ n (1 )/ 1 1/ 1/
lim
x
x n n n n
lim lim ln
ln
1
x
x
x n n
n
x
6
6
17,000,000 17 10 1/10 17
e e e
x x o x for any positive integer n
(b) ln 17,000,000 24,154,952.75
15
(c) x 3.430631121 10
15 15
(d) In the interval 3.41 10 ,3.45 10 we have
ln x 10ln(ln x ). The graphs cross at about
15
3.4306311 10 .
22.
lim
ln x
lim
1/ x
n
ln x
x x x nx n 1
1
n n 1
a
1 1 0 lim n 1
a
1
a
0 a n
n
n n a
x
n n
x x
x
n 1
x
n
lim lim 0 ln x grows slower
x
a x a x a x a a nx
than any non-constant polynomial ( n 1)
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2014 Pearson Education, Inc.