# How do you find lim (sqrt(x+1)-1)/(sqrt(x+4)-2) as x->0 using l'Hospital's Rule or otherwise?

Mar 7, 2017

$2$

#### Explanation:

${\lim}_{x \to 0} \frac{\sqrt{x + 1} - 1}{\sqrt{x + 4} - 2}$

$\frac{\sqrt{x + 1} - 1}{\sqrt{x + 4} - 2} = \frac{\left(\sqrt{x + 1} - 1\right) \left(\sqrt{x + 4} + 2\right)}{x} =$

$= \frac{\sqrt{x + 4} + 2}{\sqrt{x + 1} + 1}$ then

${\lim}_{x \to 0} \frac{\sqrt{x + 1} - 1}{\sqrt{x + 4} - 2} = {\lim}_{x \to 0} \frac{\sqrt{x + 4} + 2}{\sqrt{x + 1} + 1} = 2$