# Limit of a Function in Indeterminate Form| Differential and Integral Calculus| Feliciano and Uy| Exercise 1.3, Problem 20|

#### $\lim\limits_{x\to 0}\left(\frac{f\left(9+x\right)-f\left(9\right)}{x}\right)$

SOLUTION:

$\lim\limits_{x\to 0}\left(\frac{f\left(9+x\right)-f\left(9\right)}{x}\right)=\lim\limits_{x\to 0}\left(\frac{\sqrt{9+x}-\sqrt{9}}{x}\right)$

Direct substitution of $x=0$ gives the indeterminate form $\frac{0}{0}$. Therefore, we proceed by rationalizing the numerator

$=\lim\limits_{x\to 0}\left(\frac{\sqrt{9+x}-3}{x}\cdot \frac{\sqrt{9+x}+3}{\sqrt{9+x}+3}\right)$

$=\lim\limits_{x\to 0}\left(\frac{9+x-9}{x\left(\sqrt{9+x}+3\right)}\right)$

$=\lim\limits_{x\to 0}\left(\frac{x}{x\left(\sqrt{9+x}+3\right)}\right)$

$=\lim\limits_{x\to 0}\left(\frac{1}{\left(\sqrt{9+x}+3\right)}\right)$

$=\left(\frac{1}{\left(\sqrt{9+0}+3\right)}\right)$

$=\frac{1}{6}\:\:\:$