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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
MCQ
If $f(x)=\left\{\begin{array}{l}\frac{k x}{|x|} \text {, if } x<0 \\ 3, \text { if } x \geq 0\end{array}\right.$ is continuous at $x=0$, then the value of $k$ is
  • ✓
    $-3$
  • B
    $0$
  • C
    $3$
  • D
    any real number
✓

Answer

Correct option: A.
$-3$
$\text {} f(x)=\left\{\begin{array}{l}\frac{k x}{|x|}, \text { if } x<0 \\ 3, \text { if } x \geq 0\end{array}\right. $
Since, $ f $ is continuous at $ x=0,$
$\Rightarrow \quad \text { L.H.L }=\text { R.H.L. }=f(0) $
$\Rightarrow \quad \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$
$\Rightarrow \quad \lim _{x \rightarrow 0^{-}} \frac{-k x}{x}=\lim _{x \rightarrow 0^{+}} 3=3 $
$\Rightarrow k=-3 $

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