Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -6}\frac{4 x^{2} + 28 x + 24}{- x^{2} - 11 x - 30}$

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: $LaTeX: \lim_{x \to -6}\frac{4 x^{2} + 28 x + 24}{- x^{2} - 11 x - 30} = \lim_{x \to -6}\frac{8 x + 28}{- 2 x - 11} = \frac{8 (-6) + 28}{- 2 (-6) - 11} = -20$