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  4. If f(9)=9, f prime(9)=4, then displaystyle lim x arrow 9 (√f(x)-3/√x-3) equals

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Q. If $f(9)=9, f^{\prime}(9)=4$, then $\displaystyle\lim _{x \rightarrow 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3}$ equals

Bihar CECEBihar CECE 2010

Solution:

$\displaystyle \lim _{x \rightarrow 9} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3}$
$=\displaystyle \lim _{x \rightarrow 9} \frac{\sqrt{f(x)} - 3}{\sqrt{x}-3} \cdot \frac{\sqrt{x}+3}{\sqrt{x}+3} \cdot \frac{\sqrt{f(x)}+3}{\sqrt{f(x)}+3}$
$=\displaystyle \lim _{x \rightarrow 9} \frac{f(x)-9}{x-9} \cdot \frac{\sqrt{x}+3}{\sqrt{f(x)}+3}$
$=\displaystyle \lim _{x \rightarrow 9} f'(x) \frac{\sqrt{x}+3}{\sqrt{f(x)}+3}$
$=f^{\prime}(9) \frac{\sqrt{9}+3}{\sqrt{f(9)}+3}$
$=4 \cdot \frac{6}{6}=4$