Question

Find the following limit.
$\underset{x\to 2}{lim}\frac{x-\sqrt{3x-2}}{x^2-4}.$

Answer 1

$\underset{x\to 2}{lim}\frac{x-\sqrt{3x-2}}{x^2-4}$

by using Rationalization method

$\underset{x\to 2}{lim}\frac{x-\sqrt{3x-2}}{x^2-4}\times \frac{x+\sqrt{3x-2}}{x+\sqrt{3x-2}}$

$\underset{x\to 2}{lim}\frac{x^2-(3x-2)}{x^2-4(x+\sqrt{3x-2})}=\underset{x\to 2}{lim}\frac{x^2-3x+2}{(x+2)(x-2)(x+\sqrt{3x-2})}$

$=\underset{x\to 2}{lim}\frac{(x-1)(x-2)}{(x-2)(x+3)(x+\sqrt{3x-2})}$

$=\frac{2-1}{(2+2)\sqrt{2+}}$

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

$=\frac{1}{4\times 4}=\frac{1}{16}$