Question

If $f\left(x\right)=\left\{\begin{array}{cc}\frac{\sin 3x}{x},& \mathrm{if}x\ne 0\\ \frac{k}{2},& \mathrm{if}x=0\end{array}\right.$is continuous at x = 0, then k is equal to _____________.

Answer 1

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

$\therefore f\left(0\right)=\underset{x\to 0}{\lim }f\left(x\right)$

$\Rightarrow \frac{k}{2}=\underset{x\to 0}{\lim }\frac{\sin 3x}{x}$

$\Rightarrow \frac{k}{2}=3\underset{x\to 0}{\lim }\frac{\sin 3x}{3x}$

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

$\Rightarrow k=6$

Thus, the value of k is 6.


If $f\left(x\right)=\left\{\begin{array}{cc}\frac{\sin 3x}{x},& \mathrm{if}x\ne 0\\ \frac{k}{2},& \mathrm{if}x=0\end{array}\right.$ is continuous at x = 0, then k is equal to ____6____.