Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -1}\frac{4 x^{2} + 12 x + 8}{9 x^{2} + 16 x + 7}$

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: $LaTeX: \lim_{x \to -1}\frac{4 x^{2} + 12 x + 8}{9 x^{2} + 16 x + 7} = \lim_{x \to -1}\frac{8 x + 12}{18 x + 16} = \frac{8 (-1) + 12}{18 (-1) + 16} = -2$