Question

Find the following limit.
$\underset{x\to 2}{lim}\frac{x^2-5x+6}{x^2-12x+20}.$

Answer 1

$\underset{x\to 2}{lim}\frac{x^2-5x+6}{x^2-12x+20}$

By using factorization method

$\underset{x\to 2}{lim}\frac{x^2-2x-3x+6}{x^2-10x-2x+20}$

$\underset{x\to 2}{lim}\frac{x(x-2)-3(x-2)}{x(x-10)-2(x-10)}$

$\underset{x\to 2}{lim}\frac{(x-3)(x-2)}{(x-2)(x-10)}$

$=\frac{2-3}{2-10}=\frac{-1}{-8}=\frac{1}{8}$.