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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Determine that value of the constant 'k' so that function $\text{f(x)}=\begin{cases}\frac{\text{kx}}{|\text{x}|},&\text{if }\text{ x}<0\\3,&\text{if }\text{ x}\geq0\end{cases}$ is continuous at x = 0.
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Answer

Given, $\text{f(x)}=\begin{cases}\frac{\text{kx}}{|\text{x}|},&\text{if }\text{ x}<0\\3,&\text{if }\text{ x}\geq0\end{cases}$
Since, the function is continuous at x = 0, therefore,
$\lim\limits_{{\text{x}}\rightarrow0^-}\text{f(x})=\lim\limits_{{\text{x}}\rightarrow0^+}\text{f(x})=\text{f}(0)$
$\Rightarrow\lim\limits_{{\text{x}}\rightarrow0}\frac{-\text{kx}}{\text{x}}=\lim\limits_{{\text{x}}\rightarrow0}3=3$
$\Rightarrow-\text{k}=3$
$\Rightarrow\text{k}=-3$

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