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

If f\left(x\right)=\sqrt{x}, find

\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}\:\:\:

 

 

 

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