Question

Evaluate the limit:
$\underset{x\to 2}{lim}\frac{x^2-x-2}{x^2-2x+\sin (x-2)}$

Answer 1

We have:
$\underset{x\to 2}{lim}\frac{x^2-x-2}{x^2-2x+\sin (x-2)}$

$=\underset{x\to 2}{lim}\frac{\left(\right(x-2\left)\right(x+1\left)\right)}{\left(\right(x(x-2))+\sin (x-2\left)\right)}$

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

Dividing numerator and denominator by $(x-2)$

$=\frac{3}{3}=1$