Question

Determine the value of the constant 'k' so that function f$\left(x\right)=\left\{\begin{array}{ll}\frac{kx}{\left|x\right|},& \mathrm{if}x<0\\ 3,& \mathrm{if}x\ge 0\end{array}\right.$ is continuous at $x=$0.

Answer 1

$\mathrm{Given},f\left(x\right)=\left\{\begin{array}{ll}\frac{kx}{\left|x\right|}& ,ifx<0\\ 3& ,ifx⩾0\end{array}\right.$

Since the function is continuous at x = 0, therefore,

$$\begin{gathered} \underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }f\left(x\right)=f\left(0\right) \\ \Rightarrow \underset{x\to 0}{\lim }\frac{-kx}{x}=\underset{x\to 0}{\lim }3=3 \\ \Rightarrow -k=3 \\ \Rightarrow k=-3 \end{gathered}$$