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  4. If f (1) = 1, f1(1) = 2, then displaystyle limx → 1 (√f(x) - 1/ √x -1) is

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Q. If $f (1) = 1$, $f^1(1) = 2$, then $\displaystyle \lim_{x \to 1} \frac{\sqrt{f(x)} - 1}{ \sqrt{x} -1}$ is

IIT JEEIIT JEE 2002Limits and Derivatives

Solution:

$\displaystyle \lim_{x \to 1} \frac{\sqrt{f\left(x\right)} -1}{\sqrt{x } -1} \left(\frac{0}{0}\right) $ form using L’ Hospital’s rule
$ = \displaystyle \lim _{x \to 1} \frac{\frac{1}{2\sqrt{f\left(x\right)}}f'\left(x\right)}{1 /2 \sqrt{x}} = \frac{f'\left(1\right)}{\sqrt{f\left(1\right)}} = \frac{2}{1} = 2 $.