sin 4x
(i) lim-
x sin 2x
Answers
Step-by-step explanation:
Multiply the numerator and denominator by
2
x
.
lim
x
→
0
sin
(
4
x
)
⋅
(
2
x
)
sin
(
2
x
)
⋅
(
2
x
)
Multiply the numerator and denominator by
4
x
.
lim
x
→
0
sin
(
4
x
)
⋅
(
2
x
)
⋅
(
4
x
)
4
x
⋅
sin
(
2
x
)
⋅
(
2
x
)
Separate fractions.
lim
x
→
0
sin
(
4
x
)
4
x
⋅
2
x
sin
(
2
x
)
⋅
4
x
2
x
Split the limit using the Product of Limits Rule on the limit as
x
approaches
0
.
lim
x
→
0
sin
(
4
x
)
4
x
⋅
lim
x
→
0
2
x
sin
(
2
x
)
⋅
lim
x
→
0
4
x
2
x
The limit of
sin
(
4
x
)
4
x
as
x
approaches
0
is
1
.
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1
⋅
lim
x
→
0
2
x
sin
(
2
x
)
⋅
lim
x
→
0
4
x
2
x
The limit of
2
x
sin
(
2
x
)
as
x
approaches
0
is
1
.
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1
⋅
1
⋅
lim
x
→
0
4
x
2
x
Move the term
4
2
outside of the limit because it is constant with respect to
x
.
1
⋅
1
⋅
4
2
lim
x
→
0
x
x
Evaluate the limit of the numerator and the limit of the denominator.
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1
⋅
1
⋅
4
2
0
0
Since
0
0
is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
lim
x
→
0
x
x
=
lim
x
→
0
d
d
x
[
x
]
d
d
x
[
x
]
Find the derivative of the numerator and denominator.
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1
⋅
1
⋅
4
2
lim
x
→
0
1
1
Split the limit using the Limits Quotient Rule on the limit as
x
approaches
0
.
1
⋅
1
⋅
4
2
(
lim
x
→
0
1
)
(
lim
x
→
0
1
)
Evaluate the limits by plugging in
0
for all occurrences of
x
.
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1
⋅
1
⋅
4
2
1
1
Simplify the answer.
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2