Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -6}\frac{5 x^{2} - 5 x - 210}{6 x^{2} + 31 x - 30}$

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: $LaTeX: \lim_{x \to -6}\frac{5 x^{2} - 5 x - 210}{6 x^{2} + 31 x - 30} = \lim_{x \to -6}\frac{10 x - 5}{12 x + 31} = \frac{10 (-6) - 5}{12 (-6) + 31} = \frac{65}{41}$