Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -2}\frac{2 x^{2} + 20 x + 32}{- 6 x^{2} - 14 x - 4}$

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: $LaTeX: \lim_{x \to -2}\frac{2 x^{2} + 20 x + 32}{- 6 x^{2} - 14 x - 4} = \lim_{x \to -2}\frac{4 x + 20}{- 12 x - 14} = \frac{4 (-2) + 20}{- 12 (-2) - 14} = \frac{6}{5}$