Question

If $f(9)=f\prime (9)=0$, then $\underset{x\to 9}{\lim }\frac{\sqrt{f(x)}-3}{\sqrt{x}-3}$ is equal to

• A. 0
• B. f(0)
• C. f'(3)
• D. f(9)
• E. 1

Answer 1

Answer: (A)

$\underset{x\to 0}{\lim }\frac{\sqrt{f(x)}-3}{\sqrt{x}-3}$ [$\frac{0}{0}$ form ]
$=\underset{x\to 9}{\lim }\frac{\frac{f^{{}^{\prime }}(x)}{2\sqrt{f(x)}}}{\frac{1}{2\sqrt{x}}}$
$=\underset{x\to 0}{\lim }\frac{\sqrt{x}{f}^{{}^{\prime }}(x)}{\sqrt{f(x)}}$
$=\frac{\sqrt{9}{f}^{{}^{\prime }}(a)}{\sqrt{f(a)}}$
$=\frac{3\times 0}{3}=0$