36. (a)
Apply L’Hôpital’s Rule
n
times to find
lim
x
→
‘
.
lim
x
→
‘
5
lim
x
→
‘
}
a
e
n
n
x
!
}5‘
Thus
e
x
grows faster than
a
n
x
n
1
a
n
2
1
x
n
2
1
1
…
1
a
1
x
1
a
0
as
x
→
‘
.
(b)
Apply L’Hôpital’s Rule
n
times to find
lim
x
→
‘
.
lim
x
→
‘
5
…
5
lim
x
→
‘
}
(ln
a
n
a
n
)
n
!
a
x
}5‘
Thus
a
x
grows faster than
a
n
x
n
1
a
n
2
1
x
n
2
1
1
…
1
a
1
x
1
a
0
as
x
→
‘
.
37. (a)
lim
x
→
‘
}
l
x
n
1/
x
n
} 5
lim
x
→
‘
5
lim
x
→
‘
}
x
1
n
/
n
} 5
0
Thus ln
x
grows slower than
x
1/
n
as
x
→
‘
for any
positive integer
n
.
(b)
lim
x
→
‘
}
ln
x
a
x
} 5
lim
x
→
‘
5
lim
x
→
‘
}
a
1
x
a
} 5
0
Thus ln
x
grows slower than
x
a
as
x
→
‘
for any number
a
.
0.
38.
lim
x
→
‘
5
lim
x
→
‘
5
lim
x
→
‘
5
0
Thus ln
x
grows slower than any nonconstant
polynomial as
x
→
‘
.
39.
Compare
n
log
2
n
to
n
3/2
as
n
→
‘
.
lim
n
→
‘
}
n
l
n
o
3
g
/2
2
n
} 5
lim
n
→
‘
}
lo
n
g
1
2
/2
n
}
5
lim
n
→
‘
5
lim
n
→
‘
5
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN

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