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  4. If f(9) = f ′(9) = 0, then displaystyle limx → 9 (√f(x) - 3/√x - 3 ) is equal to

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Q. If $f(9) = f ′(9) = 0$, then $\displaystyle\lim_{x \to 9} \frac{\sqrt{f(x)} - 3}{\sqrt{x} - 3 }$ is equal to

KEAMKEAM 2017Continuity and Differentiability

Solution:

$\displaystyle \lim _{x \rightarrow 0} \frac{\sqrt{f(x)}-3}{\sqrt{x}-3}$ [$\frac{0}{0}$ form ]
$=\displaystyle \lim _{x \rightarrow 9} \frac{\frac{f^{'}(x)}{2 \sqrt{f( x)}}}{\frac{1}{2 \sqrt{x}}}$
$=\displaystyle \lim _{x \rightarrow 0} \frac{\sqrt{x} f^{'}(x)}{\sqrt{f(x)}}$
$=\frac{\sqrt{9} f^{'}(a)}{\sqrt{f(a)}}$
$=\frac{3 \times 0}{3}=0$