Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -4}\frac{2 x^{2} + 4 x - 16}{7 x^{2} + 31 x + 12}$

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: $LaTeX: \lim_{x \to -4}\frac{2 x^{2} + 4 x - 16}{7 x^{2} + 31 x + 12} = \lim_{x \to -4}\frac{4 x + 4}{14 x + 31} = \frac{4 (-4) + 4}{14 (-4) + 31} = \frac{12}{25}$