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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.
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Solution
If \[f\left( x \right)\] is continuous at
\[\Rightarrow \lim_{x \to 0} \frac{x}{\sin 3x} = k\]
\[ \Rightarrow \lim_{x \to 0} \frac{1}{\frac{\sin 3x}{x}} = k\]
\[ \Rightarrow \lim_{x \to 0} \frac{1}{\frac{3 \sin 3x}{3x}} = k\]
\[ \Rightarrow \frac{1}{3}\left( \frac{1}{\lim_{x \to 0} \frac{\sin 3x}{3x}} \right) = k\]
\[ \Rightarrow k = \frac{1}{3}\]
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