Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 9}\frac{5 x^{2} - 25 x - 180}{8 x^{2} - 77 x + 45}$

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: $LaTeX: \lim_{x \to 9}\frac{5 x^{2} - 25 x - 180}{8 x^{2} - 77 x + 45} = \lim_{x \to 9}\frac{10 x - 25}{16 x - 77} = \frac{10 (9) - 25}{16 (9) - 77} = \frac{65}{67}$