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
Answer & explanation
Correct answer: option 3
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.