Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 9}\frac{4 x^{2} - 40 x + 36}{x^{2} - 15 x + 54}$

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: $LaTeX: \lim_{x \to 9}\frac{4 x^{2} - 40 x + 36}{x^{2} - 15 x + 54} = \lim_{x \to 9}\frac{8 x - 40}{2 x - 15} = \frac{8 (9) - 40}{2 (9) - 15} = \frac{32}{3}$