The value of f ( 0 ) , so that the function f(x) = 2x - ^ -1 x 2x +...

Since, $x, \sin^{- 1}x, \tan^{- 1}x$ are continuous functions, so the
function $f$ is clearly continuous at each point of its domain except possibly at $x = 0$.

So, for $f$ to be continuous at $x = 0$

$f\left(0\right) = \lim_{x \to 0}f\left(x\right) = \lim_{x \to 0}\dfrac{2 - \dfrac{\sin^{- 1}x}{x}}{2 + \dfrac{\tan^{- 1}x}{x}} = \dfrac{1}{3}$

\left[∵ \lim_{x \to 0}\dfrac{\sin^{- 1}x}{x} = 1, \lim_{x \to 0}\dfrac{\tan^{- 1}x}{x} = 1\right]