Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -6}\frac{5 x^{2} + 75 x + 270}{- 6 x^{2} - 42 x - 36}$

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: $LaTeX: \lim_{x \to -6}\frac{5 x^{2} + 75 x + 270}{- 6 x^{2} - 42 x - 36} = \lim_{x \to -6}\frac{10 x + 75}{- 12 x - 42} = \frac{10 (-6) + 75}{- 12 (-6) - 42} = \frac{1}{2}$