Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 9}\frac{4 x^{2} - 72 x + 324}{5 x^{2} - 39 x - 54}$

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: $LaTeX: \lim_{x \to 9}\frac{4 x^{2} - 72 x + 324}{5 x^{2} - 39 x - 54} = \lim_{x \to 9}\frac{8 x - 72}{10 x - 39} = \frac{8 (9) - 72}{10 (9) - 39} = 0$