Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 9}\frac{4 x^{2} - 52 x + 144}{- 2 x^{2} + 17 x + 9}$

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: $LaTeX: \lim_{x \to 9}\frac{4 x^{2} - 52 x + 144}{- 2 x^{2} + 17 x + 9} = \lim_{x \to 9}\frac{8 x - 52}{17 - 4 x} = \frac{8 (9) - 52}{17 - 4 (9)} = - \frac{20}{19}$