Q101 / 160 · AP EAMCET 2024 (21 May Shift 2) · Mathematics

If the function $f(x)=\frac{\sqrt{1+x}-1}{x}$ is continuous at $x=0$ then $f(0)=$

  • A
    $-\frac{1}{2}$
  • B
    $\frac{1}{3}$
  • Correct answer
    C
    $\frac{1}{2}$
  • D
    $-\frac{1}{3}$

Solution

$\begin{aligned} & \text { } \lim _{x \rightarrow 0} f(x)=f(0) \Rightarrow f(0)=\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x} \\ & =\lim _{x \rightarrow 0} \frac{1+x-1}{x(\sqrt{1+x}+1)}=\lim _{x \rightarrow 0} \frac{1}{1+\sqrt{1+x}}=\frac{1}{2}\end{aligned}$

Hence, the correct answer is option: 'C'.

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