For what value of k, the function given below is continuous at x=… - Vidyadip

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
$0$
✓
$\frac{1}{4}$
C
$1$
D
$4$

✓

Answer

Correct option: B.

$\frac{1}{4}$

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

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