Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -7}\frac{4 x^{2} + 64 x + 252}{4 x^{2} + 30 x + 14}$

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: $LaTeX: \lim_{x \to -7}\frac{4 x^{2} + 64 x + 252}{4 x^{2} + 30 x + 14} = \lim_{x \to -7}\frac{8 x + 64}{8 x + 30} = \frac{8 (-7) + 64}{8 (-7) + 30} = - \frac{4}{13}$