Differential and Integral Calculus by Feliciano and Uy, Exercise 1.3, Problem 19

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PROBLEM:

If $f\left(x\right)=\sqrt{x}$ , find $\displaystyle \lim\limits_{x\to 4}\left(\frac{f\left(x\right)-f\left(4\right)}{x-4}\right)$

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SOLUTION:

\displaystyle \lim\limits_{x\to 4}\left(\displaystyle \frac{f\left(x\right)-f\left(4\right)}{x-4}\right)=\lim\limits_{x\to 4}\left(\displaystyle \frac{\sqrt{x}-\sqrt{4}}{x-4}\right)

Direct substitution of $x=4$ gives the indeterminate form $\frac{0}{0}$ . Therefore, we proceed by rationalizing the numerator.

\begin{align*} & =\lim\limits_{x\to 4}\left(\displaystyle \frac{\sqrt{x}-2}{x-4}\right)\cdot \displaystyle \frac{\sqrt{x}+2}{\sqrt{x}+2} \\ \\ & =\lim\limits_{x\to 4}\left(\displaystyle \frac{x-4}{\left(x-4\right)\left(\sqrt{x}+2\right)}\right) \\ \\ & =\lim\limits_{x\to 4}\left(\displaystyle \frac{1}{\sqrt{x}+2}\right) \\ \\ & =\left(\displaystyle \frac{1}{\sqrt{4}+2}\right) \\ \\ & =\displaystyle \frac{1}{4} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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