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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
MCQ
For what value of $k$, the function given below is continuous at $x=0$ ?
$f(x)=\left\{\begin{array}{cc}\frac{\sqrt{4+x}-2}{x} & , x \neq 0 \\ k & , x=0\end{array}\right.$
  • A
    $0$
  • B
    $\frac{1}{4}$
  • C
    1
  • D
    4
✓

Answer

$\begin{array}{l}\text {As, } f(x)=\left\{\begin{array}{cc}\frac{\sqrt{4+x-2}}{x}, & x \neq 0 \\ k, & x=0\end{array} \text { is continuous at } x=0\right. \\ \Rightarrow \quad L H L=R H L=f(0) \text { or } \lim _{x \rightarrow 0} f(x)=f(0) \\ \Rightarrow \quad \lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x} \times \frac{\sqrt{4+x}+2}{\sqrt{4+x}+2}=k \\ \Rightarrow \quad \lim _{x \rightarrow 0} \frac{4+x-4}{x(\sqrt{4+x}+2)}=k \Rightarrow k=\lim _{x \rightarrow 0} \frac{1}{(\sqrt{4+x}+2)} \\ \therefore \quad k=\frac{1}{4}\end{array}$

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