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Attempt following question by selecting a choice to answer.
D
Evaluate :
lim
x
→
∞
(1  sin
5
x
)
^{x}
f D
A.
D
∞
B.
D
1
C.
D
e
^{ 5}
D.
D
 5
E.
D
0
D
C
View Solution Steps
Step 1:
lim
x
→
∞
(1  sin
5
x
)
^{x}
[Indeterminate form of the type 1
^{∞}
.]
Step 2: Let
y
= (1  sin
5
x
)
^{x}
Step 3:
lim
x
→
∞
ln
y
=
lim
x
→
∞
x
ln(1  sin
5
x
)[Apply natural logarithm on both sides.]
Step 4: =
lim
x
→
∞
ln
(
1

sin
5
x
)
1
x
[The limit is of the form
0
0
.]
Step 5: =
lim
x
→
∞
1
1

sin
5
x
(

cos
5
x
)
(

5
x
2
)

1
x
2
[Use L'Hospital's Rule.]
Step 6: =
lim
x
→
∞

5
cos
5
x
1

sin
5
x
=  5[Evaluate the limit.]
Step 7:
lim
x
→
∞
ln
y
=  5
Step 8:
lim
x
→
∞
y
=
e
^{ 5}
Step 9: Hence
lim
x
→
∞
( 1  sin
5
x
)
^{x}
=
e
^{ 5}
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