Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 9}\frac{4 x^{2} - 324}{- 7 x^{2} + 59 x + 36}$

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: $LaTeX: \lim_{x \to 9}\frac{4 x^{2} - 324}{- 7 x^{2} + 59 x + 36} = \lim_{x \to 9}\frac{8 x}{59 - 14 x} = \frac{8 (9)}{59 - 14 (9)} = - \frac{72}{67}$