Question:
Determine the value of the constant ' $k$ ' so that function $f(x)=\left\{\begin{array}{cl}\frac{k x}{|x|}, & \text { if } x<0 \\ 3, & \text { if } x \geq 0\end{array}\right.$ is continuous at $x=0 .$
Solution:
Given, $f(x)= \begin{cases}\frac{k x}{|x|} & , \text { if } x<0 \\ 3 & , \text { if } x \geqslant 0\end{cases}$
Since the function is continuous at $x=0$, therefore,
$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$
$\Rightarrow \lim _{x \rightarrow 0} \frac{-k x}{x}=\lim _{x \rightarrow 0} 3=3$
$\Rightarrow-k=3$
$\Rightarrow k=-3$