he value of f(0) so that the function $f(x)=\frac{2x-\sin^{-1}x}{2x+\tan^{-1}x}$ is continuous at each point on its domain is

$\frac{1}{3}$

$\underset{x→0}{\lim}f(x)=f(0)$   $⇒\underset{x→0}{\lim}\frac{2x-\sin^{-1}x}{2x+\tan^{-1}x}$

$\underset{x→0}{\lim}\frac{2-\frac{\sin^{-1}x}{x}}{2+\frac{\tan^{-1}x}{x}}=\frac{2-1}{2+1}=\frac{1}{3};\,∴f(0)=\frac{1}{3}$