Rational limit by factoring

Text

Compute $\displaystyle\lim_{x\to 2}\frac{x^{2}-4}{x^{2}-3x+2}$.

Solution

Substituting $x=2$ gives the indeterminate form $\tfrac00$. Factor numerator and denominator: $\frac{x^{2}-4}{x^{2}-3x+2}=\frac{(x-2)(x+2)}{(x-2)(x-1)}=\frac{x+2}{x-1}\qquad(x\ne 2).$ Now the limit is found by substitution: $\lim_{x\to 2}\frac{x+2}{x-1}=\frac{4}{1}=4.$ $\boxed{\,L=4\,}$