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# If $f(1)=1,f'(1)=2$ then $\lim\limits\large\frac{\sqrt{f(x)-1}}{\sqrt x-1}$ is

$(a)\;2\qquad(b)\;4\qquad(c)\;1\qquad(d)\;\large\frac{1}{2}$

$\lim\limits_{x\to 1}\large\frac{\sqrt{f(x)-1}}{\sqrt{x}-1}\qquad\frac{0}{0}$ [from using L Hospital's rule]
$\Rightarrow \lim\limits_{x\to 1}\large\frac{1/2\sqrt{f(x)}f'(x)}{1/2\sqrt x}$
$\Rightarrow \large\frac{f'(1)}{\sqrt{f(1)}}$
$\Rightarrow \large\frac{2}{1}$
$\Rightarrow 2$
Hence (a) is the correct answer.