Question

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

Options

• 2

• `1/3`

• `2/3`

• `-1/3`

Answer 1

If the function f(x) = `(2x - sin^-1x)/(2x + tan^-1x)`, (x ≠ 0) is continuous at each point of its domain, then the value of f(0) is `1/3`.

Explanation:

Since, f(x) is continuous at each point of its domain.

∴ it is continuous at x = 0.

∴ f(0) = `lim_(x -> 0) "f"(x)`

= `lim_(x -> 0) ((2x - sin^-1x)/(2x + tan^-1x))`

Applying L'Hospital rule on R.H.S., we get

f(0) = `lim_(x -> 0) ((2 - 1/sqrt(1 - x^2)))/((2 + 1/(1 + x^2))`

= `(2 - 1)/(2 + 1)`

= `1/3`