# How do you evaluate the limit (sqrtx-2)/(x-4) as x approaches 4?

Nov 23, 2016

${\lim}_{x \rightarrow 4} \frac{\sqrt{x} - 2}{x - 4} = \frac{1}{4}$

#### Explanation:

The denominator can be viewed as the difference of two squares, so we can write:

${\lim}_{x \rightarrow 4} \frac{\sqrt{x} - 2}{x - 4} = {\lim}_{x \rightarrow 4} \frac{\sqrt{x} - 2}{{\sqrt{x}}^{2} - {2}^{2}}$
$\therefore {\lim}_{x \rightarrow 4} \frac{\sqrt{x} - 2}{x - 4} = {\lim}_{x \rightarrow 4} \frac{\sqrt{x} - 2}{\left(\sqrt{x} - 2\right) \left(\sqrt{x} + 2\right)}$
$\therefore {\lim}_{x \rightarrow 4} \frac{\sqrt{x} - 2}{x - 4} = {\lim}_{x \rightarrow 4} \frac{1}{\left(\sqrt{x} + 2\right)}$
$\therefore {\lim}_{x \rightarrow 4} \frac{\sqrt{x} - 2}{x - 4} = \frac{1}{\left(\sqrt{4} + 2\right)}$
$\therefore {\lim}_{x \rightarrow 4} \frac{\sqrt{x} - 2}{x - 4} = \frac{1}{\left(2 + 2\right)}$
$\therefore {\lim}_{x \rightarrow 4} \frac{\sqrt{x} - 2}{x - 4} = \frac{1}{4}$

This is confirmed visually from the graph $y = \frac{\sqrt{x} - 2}{x - 4}$
graph{(sqrt(x)-2)/(x-4) [-0.166, 4.835, -1.01, 1.49]}