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lim(x to 0) (sin^(2) x)/(sqrt2 - sqrt(1+...

` lim_(x to 0) (sin^(2) x)/(sqrt2 - sqrt(1+cos x))` equals

A

`4sqrt2`

B

`sqrt2`

C

`2 sqrt2`

D

4

Text Solution

Verified by Experts

The correct Answer is:
A
Given limit is `underset(x to 0) lim(sin^(2)x)/(sqrt2)-sqrt(1+cos x))" "[0/0" form"]`
`=underset(x to0)lim(sin^(2)x)/(sqrt2-sqrt2 cos. x/2)" "[:' 1- cos x = 2 cos^(2) .x/2]`
` =underset(x to 0)lim (sin^(2)x)/(sqrt2(1-cos.x/2))`
`underset(x to 0) lim (sin^(2)x)/(sqrt2 xx 2 sin^(2)(x/4))" "[:' 1 - cos.x/2 = 2 sin^(2). x/4]`
` underset(x to 0) limx^(2)/(2sqrt2(x/4)^(2))=16/(2sqrt2) = 4 sqrt2" "[underset(x to 0)lim sin x = underset(x to 0) lim x]`
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