Question

Find the following limit.
$\underset{x\to 3}{lim}\frac{x^2+5x+6}{x^2-8x+15}.$

Answer 1

$\underset{x\to 3}{lim}\frac{x^2-5x+6}{x^2+8x+15}$

By using factorization method

$\underset{(x\to 3}{lim}\frac{x^2-2x-3x+6}{x^2-3x-5x+15}$

$\underset{x\to 3}{lim}\frac{x(x-3)-3(x-2)}{x(x-3)-5(x-3)}$

$\underset{x\to 3}{lim}\frac{(x-3)(x-2)}{(x-5)(x-3)}$

$\underset{x\to 3}{lim}\frac{x-2}{x-5}$

$=\frac{3-2}{3-5}=\frac{-1}{-2}=\frac{1}{2}$.