Question

If $ f $ be a function such that $ f(9)=9 $ and $ f(9)=3, $ then $ \underset{x\to 9}{\mathop{\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}{\mathop{\lim }}\,\frac{\sqrt{f(x)-3}}{\sqrt{x}-3}=\underset{x\to 9}{\mathop{\lim }}\,\frac{f(x)-9}{\sqrt{f(x)+3}}.\frac{\sqrt{x}+3}{x-9} $
$ =\underset{x\to 9}{\mathop{\lim }}\,\frac{f(x)-9}{x-9}\underset{x\to 9}{\mathop{\lim }}\,\frac{\sqrt{x}+3}{\sqrt{f(x)}+3} $
$ =f'(9)\frac{\sqrt{9}+3}{\sqrt{f(9)}+3} $
$ =3\times \left( \frac{3+3}{\sqrt{9}+3} \right)=3z $