Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -9}\frac{5 x^{2} + 15 x - 270}{8 x^{2} + 79 x + 63}$

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: $LaTeX: \lim_{x \to -9}\frac{5 x^{2} + 15 x - 270}{8 x^{2} + 79 x + 63} = \lim_{x \to -9}\frac{10 x + 15}{16 x + 79} = \frac{10 (-9) + 15}{16 (-9) + 79} = \frac{15}{13}$