Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -7}\frac{3 x^{2} + 33 x + 84}{x^{2} + 12 x + 35}$

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: $LaTeX: \lim_{x \to -7}\frac{3 x^{2} + 33 x + 84}{x^{2} + 12 x + 35} = \lim_{x \to -7}\frac{6 x + 33}{2 x + 12} = \frac{6 (-7) + 33}{2 (-7) + 12} = \frac{9}{2}$