Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -6}\frac{4 x^{2} + 44 x + 120}{- 3 x^{2} - 17 x + 6}$

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: $LaTeX: \lim_{x \to -6}\frac{4 x^{2} + 44 x + 120}{- 3 x^{2} - 17 x + 6} = \lim_{x \to -6}\frac{8 x + 44}{- 6 x - 17} = \frac{8 (-6) + 44}{- 6 (-6) - 17} = - \frac{4}{19}$