Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -4}\frac{3 x^{2} + 36 x + 96}{- 5 x^{2} - 17 x + 12}$

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: $LaTeX: \lim_{x \to -4}\frac{3 x^{2} + 36 x + 96}{- 5 x^{2} - 17 x + 12} = \lim_{x \to -4}\frac{6 x + 36}{- 10 x - 17} = \frac{6 (-4) + 36}{- 10 (-4) - 17} = \frac{12}{23}$