Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -4}\frac{3 x^{2} + 21 x + 36}{7 x^{2} + 23 x - 20}$

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: $LaTeX: \lim_{x \to -4}\frac{3 x^{2} + 21 x + 36}{7 x^{2} + 23 x - 20} = \lim_{x \to -4}\frac{6 x + 21}{14 x + 23} = \frac{6 (-4) + 21}{14 (-4) + 23} = \frac{1}{11}$