If the function $f(x) =\left\{\begin{matrix}\frac{\sin 3x}{x},&if\,x≠0\\\frac{3k}{2},&if\,x=0\end{matrix}\right.$ is continuous at $x = 0$, then the value of $k$ is

2

The correct answer is Option (3) → 2

Given:

$f(x)=\frac{\sin 3x}{x}$ for $x\neq 0$

$f(0)=\frac{3k}{2}$

Continuity at $x=0$ requires:

$\lim_{x\to 0}\frac{\sin 3x}{x}=f(0)$

$\lim_{x\to 0}\frac{\sin 3x}{x} =\lim_{x\to 0}3\frac{\sin 3x}{3x}=3$

Thus:

$\frac{3k}{2}=3$

$3k=6$

$k=2$

The value of $k$ is 2.