If the function \(f(x) = \begin{cases}\displaystyle\frac{\sin 3x}{x}, & \text{if } x \ne 0\\ \displaystyle\frac{3k}{2}, & \text{if } x = 0\end{cases}\) is continuous at \(x = 0\), then the value of \(k\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2
f is continuous at x = 0 when the limit of f(x) as x → 0 equals f(0).
Using the standard limit \(\lim_{\theta\to 0}\frac{\sin\theta}{\theta} = 1\) with \(\theta = 3x\):
\( \displaystyle \lim_{x\to 0}\frac{\sin 3x}{x} = \lim_{x\to 0} 3\cdot\frac{\sin 3x}{3x} \)
\(= 3 \times 1 = 3\)
Equating with f(0):
\( \displaystyle \frac{3k}{2} = 3 \)
\(k = 2\)