Question

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

• A. 9
• B. 3
• C. 1
• D. 6
• E. $\frac{2}{3}$

Answer 1

Answer: (B)

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