Question

If $ f $ be a function such that $ f (9) =9 $ and $ f' (9) = 3 $ , then $ \displaystyle\lim _{x\to9} \frac{\sqrt{f \left(x\right)-3}}{\sqrt{x}-3} $ is equal to

• A. $ 9 $
• B. $ 3 $
• C. $ 1 $
• D. None of these

Answer 1

Answer: (B)

$\displaystyle\lim_{x\to9} \frac{\sqrt{f \left(x\right)}}{\sqrt{x}-3}$
$=\displaystyle\lim_{x\to9} \frac{f \left(x\right)-9}{x-9}\times\frac{\sqrt{x}+3}{\sqrt{f \left(x\right)}+3}$
$=f' \left(9\right)\times\left(\frac{\sqrt{9}+3}{\sqrt{f\left(9\right)}+3}\right)$
$=f'\left(9\right)\times\frac{3+3}{3+3}$
$=f'\left(9\right)$
$=3$