The function


The function $f(x)=\left\{\begin{array}{ll}\frac{\sin 3 x}{x}, & x \neq 0 \\ \frac{k}{2} & , x=0\end{array}\right.$ is continuous at $x=0$, then $k=$

(a) 3

(b) 6

(c) 9

(d) 12


(b) 6

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

If $f(x)$ is continuous at $x=0$, then

$\lim _{x \rightarrow 0} f(x)=f(0)$

$\Rightarrow \lim _{x \rightarrow 0} \frac{\sin 3 x}{x}=f(0)$

$\Rightarrow 3 \lim _{x \rightarrow 0} \frac{\sin 3 x}{3 x}=\frac{k}{2}$

$\Rightarrow 3 \times 1=\frac{k}{2}$

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

$\Rightarrow k=6$

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