Thomas Calculus 13th [Solutions]

Chapter 7 Practice Exercises 537
(d)
(e)
1 1
x
1
x
2
1 1
x
x
1 x
2
1
1 2
1
1
x
x
x
2
lim
tan
tan
lim lim lim 1
x x x x
x
2
1 1 1
2
1 1 sin x
1 x
x
x
2 3
1
x
2x
2 1
1
2
x
2
x
lim
sin
lim lim lim
x x x x
same rate
faster
(f )
x
e
x
2
e
x
x x x x x
2x
sech
lim lim lim 2 lim 2 2
x e x e x e e e x 1 e
same rate
21. (a)
(b)
1 1
x
2
x
4
1
x
2
1 1
x
2
x
4
1
x
4
1 1 2 for x sufficiently large true
2
x
2
x 1 M for any positive integer M whenever x M false
(c) lim x lim 1 1
ln 1
x
x x
x 1
(d)
(e)
(f )
x
1/ x
ln x
lim ln(ln x) 1
ln x
lim lim ln x
0
x x x
1
1
x
the same growth rate false
tan x for all x true
1 2
cosh x 1 2x
1 e
1 (1 1) 1 if x 0 true
x
e 2 2
grows slower true
22. (a)
1
x
4
1 1
x
2
x
4
1
2
x 1
1
if x 0 true
(b)
1
x
4
lim lim 1 0
2
x
x x 1
1 1
x
2
x
4
1
x
(c) lim ln x lim 0
x
x 1
x
1
(d) ln 2x
ln 2 1 1 1 2
ln x ln x
(e)
(f )
true
true
if x 2 true
1 1
1 cos
x x 2
if 1
sec
1 1 1 2
x true
sinh x 1 2x
1 e
1 if x 0 true
x
e 2 2
23.
1 1
1
df x
df x df 1 df
e
1 1 1
dx dx dx 1 2 1 3
x f (ln 2) x f (ln 2) e
dx
x ln 2
x ln 2
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