Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -4}\frac{4 x^{2} - 16 x - 128}{3 x^{2} + 14 x + 8}$

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: $LaTeX: \lim_{x \to -4}\frac{4 x^{2} - 16 x - 128}{3 x^{2} + 14 x + 8} = \lim_{x \to -4}\frac{8 x - 16}{6 x + 14} = \frac{8 (-4) - 16}{6 (-4) + 14} = \frac{24}{5}$