Thomas Calculus 13th [Solutions]

532 Chapter 7 Integrals and Transcendental Functions
8.
x
x
2 x
2
(ln 2) ln(ln 2) (ln 2) ln(ln 2) (ln 2) ln(ln 2)
2
x x x
2x
x
2 2
x
lim lim lim lim (ln 2) 0 (ln 2)
x 2 ;
2
x
2x
2
x 2 x (ln 2)2 x (ln 2) 2
x
lim lim lim 0
x x 2 x
2
x grows slower than 2 ;
x
x 2x
x
2
x
x e x
e
x 2 x x
x
x
grows slower than
lim lim 0 2
grows slower than e . Therefore, the slowest to the fastest is: ln 2 , , 2 and e so the order is c, b, a, d
x
9. (a) false; lim
x
1
x
x
(b) false; lim
x 1
1
(c) true;
(d) true;
x
x 5 1
x
x 5
x
2x
x
e 1
x e x
x
e
x x x x
x x x x
x x 5 1 if x 1 (or sufficiently large)
x 2x 1 if x 1 (or sufficiently large)
(e) true; lim lim 0
2x
(f ) true;
ln
1
ln
1 1
1
2
if x 1 (or sufficiently large)
(g) false;
(h) true;
1
x
2
2x
ln x
x
ln 2x
x x
lim lim lim 1 1
2
x 5 ( x 5) 2
x 5 5
x x x x
1 6
if x 1 (or sufficiently large)
10. (a) true;
1
x 3 x
1 x 3
x
1
if x 1 (or sufficiently large)
(b) true;
1 1
x
x
2
1
x
1
x
1 2
if x 1 (or sufficiently large)
(c) false;
x
1 1
x
x
2
1
x
1
x
lim lim 1 1
x
(d) true; 2 cos x 3
2 cos x 3
if x is sufficiently large
x
x
e x x
x
e e
(e) true; 1
(f ) true;
2 2
and
x
0 as x 1
x
2 if x is sufficiently large
e
x
x
lim
xln
x
lim
ln x
lim 0
2
x x x
x
x
1
x x
1 if x is sufficiently large
ln x ln x
(g) true; ln(ln ) ln
(h) false;
1
x
x
1
x
2
x
2
2x
x
2 1
2 2
lim
ln
lim lim
1
lim
1 1 1
x ln( x 1) x x 2x x
2 2x
2
f ( x) g( x) 11. If f ( x ) and g( x ) grow at the same rate, then lim 0 lim
1
f ( x)
L 0. Then L 1 if x is
x
g( x) x
f ( x)
L
g( x)
( ) ( )
sufficiently large L f x
f x
1 1 1
g( x) L g( x)
L if x is sufficiently large f O ( g ). Similarly,
g( x) 1
1 g O( f ).
f ( x)
L
( )
12. When the degree of f is less than the degree of g since in that case lim f x
g( x)
0.
x
e
Copyright
2014 Pearson Education, Inc.