Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -9}\frac{5 x^{2} + 80 x + 315}{- 3 x^{2} - 22 x + 45}$

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: $LaTeX: \lim_{x \to -9}\frac{5 x^{2} + 80 x + 315}{- 3 x^{2} - 22 x + 45} = \lim_{x \to -9}\frac{10 x + 80}{- 6 x - 22} = \frac{10 (-9) + 80}{- 6 (-9) - 22} = - \frac{5}{16}$