Question

Determine the value of the constant 'k' so that function f 

\[\left( x \right) = \begin{cases}\frac{kx}{\left| x \right|}, &\text{ if }  x < 0 \\ 3 , & \text{ if } x \geq 0\end{cases}\]  is continuous at x  = 0  . 

Answer 1

\[\text{ Given } , f(x) = \begin{cases}\frac{kx}{\left| x \right|} & , \text{ if } x < 0 \\ 3 & , \text{ if } x \geqslant 0\end{cases}\]

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

\[\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)\]
\[ \Rightarrow \lim_{x \to 0} \frac{- kx}{x} = \lim_{x \to 0} 3 = 3\]
\[ \Rightarrow - k = 3\]
\[ \Rightarrow k = - 3\]