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 $x=0$

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

$\left[\because \underset{x\to 0}{\lim }\frac{{\sin }^{-1}x}{x}=1,\underset{x\to 0}{\lim }\frac{{\tan }^{-1}x}{x}=1\right]$