Question

If $f\left(1\right)=1$, $f^{\prime }\left(1\right)=2$ then $\underset{x\to 1}{lim}\frac{\sqrt{f\left(x\right)}-1}{\sqrt{x}-1}$ equals

• A. $2$
• B. $4$
• C. $1$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $2$

$\underset{x\to 1}{lim}\frac{\sqrt{f\left(x\right)}-1}{\sqrt{x}-1}$ (0/0 form)
$\underset{x\to 1}{lim}\frac{1}{2\sqrt{f\left(x\right)}}\times \frac{2\sqrt{x}}{1}\times f^{\prime }\left(x\right)$
$=\frac{2\times 1\times 2}{2}=2$