Answers and Explanations 497
Answers
801–900
Applying the limit gives you the indeterminate form ∞
0
. Create a new limit by taking
the natural logarithm of the original expression, y x
x
=
1
2
, so that ln
ln
y
x
x
= ( )
1
2
, or
ln
ln
y
x
x
= 2 . Then take the limit of this new expression:
lim ln
lim ln
x
x
y
x
x
→∞
→∞
=
2
Apply L’Hôpital’s rule:
lim ln
lim
lim
x
x
x
x
x
x
x
x
→∞
→∞
→∞
=
=
=
2
2
1
2
1
2
0
Because x
y
x
1
2 = and y e
y
=
ln , it follows that
lim
lim
lim
ln
lim ln
x
x
x
x
y
y
x
y
e
e
e
x
→∞
→∞
→∞
=
=
=
=
=
→∞
0
1
2
1
823.
1
Here’s the given problem:
lim cos
x
x
x
→
+
(
)
0
2
Applying the limit gives you the indeterminate form 1
∞
. Create a new limit by taking the
natural logarithm of the original expression, y = (cos x)
2/x
, so that ln y = ln(cos x)
2/x
, or
ln
ln cos
y
x
x
=
(
)
2
. Then take the limit of this new expression:
lim ln
lim
ln cos
x
x
y
x
x
→
→
+
+
=
(
)
0
0
2
Apply L’Hôpital’s rule:
lim
ln cos
lim
cos
sin
tan
x
x
x
x
x
x
→
→
+
+
(
) =
( ) −
(
)
= −
( )
=
0
0
2
2
1
1
2
0
0
Because cos x
y
x
(
) =
2
and y e
y
=
ln , it follows that
lim cos
lim
lim
ln
lim ln
x
x
x
x
y
y
x
y
e
e
e
x
→
→
→
+
+
+
→ +
(
) =
=
=
=
=
0
2
0
0
0
0
1
824.
e
−15/9
The given problem is
lim
x
x
x
x
→∞
−
−
+
( )
3 1
3
4
1
Applying the limit gives you the indeterminate form 1
∞
. Create a new limit by taking the
natural logarithm of the original expression, y
x
x
x
=
−
+
( )
−
3 1
3
4
1
, so that ln
ln
y
x
x
x
=
−
+
( )
−
3 1
3
4
1
,