Question

Evaluate : $\underset{x\to 3}{lim}\frac{x^2-x-6}{x^3-{3x}^2+x-3}$

Answer 1

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

$\underset{x\to 3}{lim}\frac{x^2-3x+2x-6}{x^2(x-3)+1(x-3)}$

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

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

$=\frac{3+2}{9+1}=\frac{5}{10}=\frac{1}{2}$.