Answers and Explanations 491
Answers
801–900
Notice that if you substitute in the value x = 1, you get the indeterminate form 0
0
.
Using L’Hôpital’s rule gives you
lim
ln
cos
lim
sin
sin
x
x
x
x
x
x
→
→
( )
=
−
( )






=
−
( )
= −
1
1
2
1
2
2
1
1
2
2
π
π
π
π
π
2 2
π
807.
1
The given problem is
lim tan
x
x
x
→
−
0
1
Notice that if you substitute in the value x = 0, you get the indeterminate form 0
0
. Using
L’Hôpital’s rule gives you the limit:
lim tan
lim
x
x
x
x
x
→
−
→
=
+
= +
=
0
1
0
2
2
1
1
1
1
1 0
1
808.
− 3
2
Here’s the given problem:
lim
x
x
x
x
→
+
−
− −






2
2
8
4
2
Applying the limit gives you the indeterminate form ∞ – ∞. Write the expression as a
single fraction by getting common denominators and then apply L’Hôpital’s rule:
lim
lim
lim
x
x
x
x
x
x
x x
x
x
x
→
→
→
+
+
+
−
− −





 =
−
+
(
)
−
=
− −
+
2
2
2
2
2
2
8
4
2
8
2
4
2
8 8
4
2
2
2
2 2 2
2 2
3
2
2
2
x
x
x
x
−
=
− −
=
− ( )−
( )
= −
→
+
lim
809.
2
The given problem is
lim
sin
x
x
x
→
( )
0
4
2