Math, asked by ramtushar1016, 4 months ago

sin 4x
(i) lim-
x sin 2x​

Answers

Answered by sushamabhopale
0

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

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