Thomas Calculus 13th [Solutions]

282 Chapter 4 Applications of Derivatives
27.
0
sin sin
3 (ln 3)(1)
lim 3 1 lim 3 (ln 3)(cos )
0 0
1 1
ln 3
28.
1 1
2 2
1
lim 1 ln
2
lim 1
0 0
1 ln 2
ln1 ln 2 ln 2
29.
x
x
x
2 (1) 2 ( x)(ln 2) 2
0
lim
1 2 0 1
x
x
0
x
lim
x 0 2 1 x 0 (ln 2) 2 (ln 2) 2
ln 2
30.
x
x
lim 3 1 lim 3 ln 3 3 ln 3 ln 3
x
x 0
x 0 2 1 x 0 2 ln 2 2 ln 2 ln 2
0
31.
1
x x x 1
x
2
ln x
1
x
x
x
x
ln 2
x
x
x x
lim ln( 1) ln( 1) 1
log lim (ln 2) lim (ln 2) lim 1 (ln 2) lim 1
ln 2
32.
ln 1
log2
x x
ln 2 ln 3 ln x ln 3 x ln 3 x 3 ln 3 1 ln 3
ln( 3)
3 x x
x 1
x
ln 3
x 3
lim lim lim lim lim lim
x
log ( 3)
x
ln 2
x
ln( 3) ln 2
x
ln 2
x
ln 2
x
1 ln 2
33.
2
2x
2
x
2 2 x
22
1
x
x
ln x 2x
lim lim lim 2x 2x lim 4x
2 lim 2 1
ln x
2
2x
2 2
x 0 x 0 x 0 x x 0 x 0
34.
x
e
x
e
x
1
1
x
ln e 1
x x x
lim lim lim xe lim e xe 1 0 1
ln x x
x
1
1
x 0 x 0 x 0 e x 0 e
35.
5y
25 5
1/2
(5y
25) 5 (5y
25) (5)
lim lim lim lim 5 1
y 0
y
y 0
y
y 0
1
y 0 2 5y
25 2
1
2
1/2
36.
2
1/2
2
1/2
1
( )
2
1
y
1 2 2
lim
2
ay a a lim ay a a lim ay a a
lim a , a 0
y 0
y
y 0 y 0 y 0 2 ay a
37. lim ln 2x
ln( x 1) lim ln 2x
ln lim 2x
ln lim 2 ln 2
x x
x 1
x
x 1
x
1
38. lim (ln x ln sin x) lim ln x ln lim x ln lim 1 ln1 0
sin x sin x cos x
x 0 x 0
x 0 x 0
39.
x
1
x
cos x
sin x
lim
2
(ln x) 2(ln )
2(ln )(sin ) 2(ln ) sin
ln(sin ) lim lim x x
cos lim x x
x x x cos x x
1
x 0 x 0 x 0 x 0
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