Question

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

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

Answer 1

Answer: (B)

$\underset{x\to 9}{\lim }\frac{\sqrt{f\left(x\right)}}{\sqrt{x}-3}$
$=\underset{x\to 9}{\lim }\frac{f\left(x\right)-9}{x-9}\times \frac{\sqrt{x}+3}{\sqrt{f\left(x\right)}+3}$
$=f^{\prime }\left(9\right)\times \left(\frac{\sqrt{9}+3}{\sqrt{f\left(9\right)}+3}\right)$
$=f^{\prime }\left(9\right)\times \frac{3+3}{3+3}$
$=f^{\prime }\left(9\right)$
$=3$