Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -2}\frac{2 x^{2} + 22 x + 36}{- 8 x^{2} - 8 x + 16}$

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: $LaTeX: \lim_{x \to -2}\frac{2 x^{2} + 22 x + 36}{- 8 x^{2} - 8 x + 16} = \lim_{x \to -2}\frac{4 x + 22}{- 16 x - 8} = \frac{4 (-2) + 22}{- 16 (-2) - 8} = \frac{7}{12}$