Question

If $f\left(x\right)=\left\{\begin{array}{ll}\frac{{\sin }^{-1}x}{x},& x\ne 0\\ k,& x=0\end{array}\right.$is continuous at x = 0, write the value of k.

Answer 1

Given, $f\left(x\right)=\left\{\begin{array}{l}\frac{{\sin }^{-1}x}{x},x\ne 0\\ k,x=0\end{array}\right.$

If $f\left(x\right)$ is continuous at $x=0$, then

$\underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right)$

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

$$\begin{gathered} \Rightarrow \underset{x\to 0}{\lim }\left(\frac{{\sin }^{-1}x}{x}\right)=k \\ \Rightarrow k=1\left[\because \underset{x\to 0}{\lim }\left(\frac{{\sin }^{-1}x}{x}\right)=1\right] \end{gathered}$$