Question

Find the following limit:
$\underset{x\to 5}{lim}\frac{\sqrt{x-1}-2}{x-5}.$

Answer 1

$\underset{x\to 5}{lim}\frac{\sqrt{x-1}-2}{x-5}$

By using Rationalization method

$\underset{x\to 5}{lim}\frac{\sqrt{x-1}-2}{x-5}\times \frac{\sqrt{x-1}+2}{\sqrt{x-1}+2}$

$\underset{x\to 5}{lim}\frac{(x-1)-4}{(x-5)(\sqrt{x-1}+2)}$

$\underset{x\to 5}{lim}\frac{x-5}{(x-5)(\sqrt{x-1}+2)}$

$\underset{x\to 5}{lim}\frac{1}{(\sqrt{x-1}+2)}$

$=\frac{1}{\sqrt{4+2}}=\frac{1}{4}$.