Question

If \[f\left( x \right) = \begin{cases}\frac{x}{\sin 3x}, & x \neq 0 \\ k , & x = 0\end{cases}\]  is continuous at x = 0, then write the value of k.

Answer 1

If  \[f\left( x \right)\]  is continuous at

\[x = 0\] , then

\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]

\[\Rightarrow \lim_{x \to 0} \frac{x}{\sin 3x} = k\]
\[ \Rightarrow \lim_{x \to 0} \frac{1}{\frac{\sin 3x}{x}} = k\]
\[ \Rightarrow \lim_{x \to 0} \frac{1}{\frac{3 \sin 3x}{3x}} = k\]
\[ \Rightarrow \frac{1}{3}\left( \frac{1}{\lim_{x \to 0} \frac{\sin 3x}{3x}} \right) = k\]
\[ \Rightarrow k = \frac{1}{3}\]