Question

The function $f\left(x\right)=\left\{\begin{array}{ll}\frac{\sin 3x}{x},& x\ne 0\\ \frac{k}{2},& x=0\end{array}\right.$ is continuous at x = 0, then k =
(a) 3
(b) 6
(c) 9
(d) 12

Answer 1

(b) 6

Given: $f\left(x\right)=\left\{\begin{array}{l}\frac{\sin 3x}{x}\\ \frac{k}{2},x=0\end{array}\right.,x\ne 0$

If $f\left(x\right)$ is continuous at $x=0$, then
$\underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right)$

$$\begin{gathered} \Rightarrow \underset{x\to 0}{\lim }\frac{\sin 3x}{x}=f\left(0\right) \\ \Rightarrow 3\underset{x\to 0}{\lim }\frac{\sin 3x}{3x}=\frac{k}{2} \\ \Rightarrow 3\times 1=\frac{k}{2} \\ \Rightarrow \frac{k}{2}=3 \\ \Rightarrow k=6 \end{gathered}$$