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Question:

If $f(x)=\left\{\begin{array}{cl}\frac{\sin 3 x}{x}, & \text { if } x \neq 0 \\ \frac{k}{2}, & \text { if } x=0\end{array}\right.$ is continuous at $x=0$, then $k$ is equal to_____________

Solution:

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

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

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

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

$\Rightarrow \frac{k}{2}=3 \times 1=3$        $\left(\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\right)$

$\Rightarrow k=6$

Thus, the value of k is 6.

If $f(x)=\left\{\begin{array}{ll}\frac{\sin 3 x}{x}, & \text { if } x \neq 0 \\ \frac{k}{2}, & \text { if } x=0\end{array}\right.$ is continuous at $x=0$, then $k$ is equal to ___6____.

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