Question

If $f\left(x\right)=\left\{\begin{array}{ll}\frac{x}{\sin 3x},& x\ne 0\\ k,& x=0\end{array}\right.$is continuous at x = 0, then write the value of k.

Answer 1

If $f\left(x\right)$ is continuous at $x=0$, then
$\underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right)$

$$\begin{gathered} \Rightarrow \underset{x\to 0}{\lim }\frac{x}{\sin 3x}=k \\ \Rightarrow \underset{x\to 0}{\lim }\frac{1}{{\displaystyle \frac{\sin 3x}{x}}}=k \\ \Rightarrow \underset{x\to 0}{\lim }\frac{1}{{\displaystyle \frac{3\sin 3x}{3x}}}=k \\ \Rightarrow \frac{1}{3}\left(\frac{1}{\underset{x\to 0}{\lim }{\displaystyle \frac{\sin 3x}{3x}}}\right)=k \\ \Rightarrow k=\frac{1}{3} \end{gathered}$$