Part II: The Answers
492
Answers
801–900
Notice that if you substitute in the value x = 0, you get the indeterminate form 0
0
.
Apply L’Hôpital’s rule to find the limit:
lim
sin
lim
cos
cos
x
x
x
x
x
→
→
( ) =
( )
=
( )
=
0
0
4
2
4
4
2
2
0
2
810.
0
Here’s the given problem:
lim ln
x
x x
→
+
0
Applying the limit gives you the indeterminate form 0(–∞). Rewrite the product as a
quotient and use L’Hôpital’s rule to find the limit:
lim ln
lim
lim
lim
x
x
x
x
x
x
x
x
x
x
x
→
−
→
−
→
→
+
+
+
+
=
−
=
− ( )
=
−
( )
=
0
1
0
2
0
2
0
1
1
0
811.
0
Here’s the given problem:
lim ln
x
x
x
→∞
4
3
Applying the limit gives you the indeterminate form ∞
∞
. Use properties of logarithms to
rewrite the expression and then use L’Hôpital’s rule to find the limit:
lim ln
lim
lim
x
x
x
x
x
x
x
x
→∞
→∞
→∞
=
=
=
4
4
3
4
3
0
3
2
3
812.
1
The given problem is
lim sin
x
x
x
→
−
0
1
Substituting the value x = 0 gives you the indeterminate form 0
0
. Apply L’Hôpital’s rule
to find the limit:
lim
x
x
→
−
=
−
=
0
2
2
1
1
1
1
1 0
1