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If f(x)=((1-tanx)/(1+sinx))^(cosec x) is...

If `f(x)=((1-tanx)/(1+sinx))^(cosec x)` is to be made continuous at `x=0,` then `f(0)` must be equal to

A

`e^(2)`

B

`e`

C

`(1)/(e)`

D

`(1)/(e^(2))`

Text Solution

Verified by Experts

The correct Answer is:
D
If f (x) is continuous at x=0,`
then underset(xrarr0)lim f (x) -f(0)`
`impliesf (0) =underset(xrarr0)lim f (x) = e^(underset(srarr0)lim((-tanx-sin x)/((1-sin x).sin x)))`
`= e^(underset(xrarr0)(lim)).((1+cos x))/((1-sin x),cos x)=e^(-2)`
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