Question

Find the following limit:
$\underset{x\to 6}{lim}\frac{\sqrt{x+3}-3}{x-6}.$

Answer 1

$\underset{x\to 6}{lim}\frac{\sqrt{x+3}-3}{x-6}$

Rationalization

$\underset{x\to 6}{lim}\frac{\sqrt{x+3}-3}{x-6}\times \frac{\sqrt{x+3}+3}{\sqrt{x+3}+3}$

$=\underset{x\to 6}{lim}\frac{(x+3)-9}{(x-6)(\sqrt{x+3}+3)}$

$=\underset{x\to 6}{lim}\frac{x-6}{(x-6)(\sqrt{x+3}+3)}$

$=\underset{x\to 6}{lim}\frac{1}{(\sqrt{x+3}+3)}$

$=\frac{1}{(\sqrt{9}+3)}=\frac{1}{6}$.