Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -7}\frac{3 x^{2} + 9 x - 84}{7 x^{2} + 58 x + 63}$

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: $LaTeX: \lim_{x \to -7}\frac{3 x^{2} + 9 x - 84}{7 x^{2} + 58 x + 63} = \lim_{x \to -7}\frac{6 x + 9}{14 x + 58} = \frac{6 (-7) + 9}{14 (-7) + 58} = \frac{33}{40}$