Part II: The Answers
494
Answers
801–900
Applying the limit gives you the indeterminate form ∞ – ∞. Rewrite the difference as a
quotient and use L’Hôpital’s rule:
lim sec
tan
lim cos
sin
cos
lim
sin
x
x
x
x
x
x
x
x
→
→
→
−
−
−
−
(
)=
−
(
)
=
−
π
π
π
2
2
2
1
1
x x
x
x
x
x
cos
lim
cos
sin
cos
sin
(
)
=
−
−
( )
=
= =
→
−
π
π
π
2
2
2
0
1
0
816.
1
Start with the given problem:
lim sin
x
x
x
→∞
( )
1
Applying the limit gives you the indeterminate form (∞)(0). Rewrite the product as a
quotient and use L’Hôpital’s rule as follows:
lim
sin
lim
cos
lim cos
x
x
x
x
x
x
x
x
x
→∞
−
→∞
→∞
( ) =
( ) ⋅ −






−
=
(
1
1
1
1
1
1
2
2
) )
=
( )
=
cos 0
1
817.
0
Here’s the initial problem:
lim
x
x
x e
→−∞
3
Taking the limit gives you the indeterminate form (–∞)(0). Rewrite the product as a
quotient and use L’Hôpital’s rule:
lim
lim
lim
x
x
x
x
x
x
x e
x
e
x
e
→−∞
→ −∞ −
→−∞
−
=
=
−
3
3
2
3
Applying the limit again gives you the indeterminate form − ∞
∞
. Using L’Hôpital’s rule
again gives you the following:
lim
lim
x
x
x
x
x
e
x
e
→−∞
−
→−∞ −
−
=
3
6
2