Question

If f(9)= 9and f'(9)= 4 then what is $\underset{x\to 9}{lim}\frac{\sqrt{f\left(x\right)}-3}{\sqrt{x}-3}$ equal to ?

• A. 36
• B. 9
• C. 4
• D. None of the above

Answer 1

The correct option is C 4

Let $L=\underset{x\to 9}{lim}\frac{\sqrt{f\left(x\right)}-3}{\sqrt{x}-3}$

The given limit reduces in zero by zero form by substituting the limit

So, using L'Hospital's rule, we get

$L=\underset{x\to 9}{lim}\frac{\frac{f^{\prime }\left(x\right)}{\sqrt{f\left(x\right)}}}{\frac{1}{\sqrt{x}}}=\frac{\frac{f^{\prime }\left(9\right)}{\sqrt{f\left(9\right)}}}{\frac{1}{\sqrt{9}}}=\frac{\frac{4}{\sqrt{9}}}{\frac{1}{\sqrt{9}}}=4$