Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -3}\frac{5 x^{2} - 20 x - 105}{- x^{2} + 6 x + 27}$

The limit is an indeterminate form of the type $LaTeX: \displaystyle \frac{0}{0}$ . Using L'Hospitial's rule and then the evaluation theorem gives: $LaTeX: \lim_{x \to -3}\frac{5 x^{2} - 20 x - 105}{- x^{2} + 6 x + 27} = \lim_{x \to -3}\frac{10 x - 20}{6 - 2 x} = \frac{10 (-3) - 20}{6 - 2 (-3)} = - \frac{25}{6}$