Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 4}\frac{3 x^{2} + 12 x - 96}{- 7 x^{2} + 24 x + 16}$

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: $LaTeX: \lim_{x \to 4}\frac{3 x^{2} + 12 x - 96}{- 7 x^{2} + 24 x + 16} = \lim_{x \to 4}\frac{6 x + 12}{24 - 14 x} = \frac{6 (4) + 12}{24 - 14 (4)} = - \frac{9}{8}$