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If the function f(x)=(2x-sin^(-1)x)/(2x+...

If the function `f(x)=(2x-sin^(-1)x)/(2x+tan^(-1)x)` is continuous at each point of its domain, then the value of `f(0)`

A

`(1)/(3)`

B

`-(1)/(3)`

C

`(2)/(3)`

D

`(-2)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
A
Since, x `sin^(-1)x, tan^(-1)x` are continuous functions, so the function f is clearly continuous at each point in its
domain except possibly at x = 0.
So, for f to be continuous at x = 0.
`therefore f(0)= lim_(x to 0) f(x)= lim_(x to 0) (2-(sin^(-1)x)/(x))/(2+(tan^(-1)x)/(x))=(1)/(3) `
`[because lim_(x to 0) (sin^(-1)x)/(x)=1, lim_(x to 0) (tan^(-1)x)/(x)=1]`
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