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If f(x)=(sqrt(1+sinx)-sqrt(1-sinx))/(x),...

If `f(x)=(sqrt(1+sinx)-sqrt(1-sinx))/(x)`, then we value of f at x = 0, so that f is continuous everywhere, is `

A

`(1)/(4)`

B

`-1`

C

1

D

2

Text Solution

Verified by Experts

The correct Answer is:
C
Since it is clear that f is continuous everywhere except possibly at x = 0, so at x = 0 for f to be continuous.
`f(0)= lim_(x to 0) f(x)`
`=lim_(x to 0) ((1+sin x)^(1//2)-(1-sin x)^(1//2))/(x)`
`=lim_(x to 0) ((1+sin x)-(1-sinx))/(x)*(1)/((1+sinx)^(1//2)+(1-sinx)^(1//2))`
`=2* lim_(x to 0) (sin x)/(x)*(1)/((1+sinx)^(1//2)+(1-sinx)^(1//2))`
`=2xx1xx(1)/((1+1))=1`
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