Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 9}\frac{3 x^{2} - 45 x + 162}{7 x^{2} - 61 x - 18}$

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: $LaTeX: \lim_{x \to 9}\frac{3 x^{2} - 45 x + 162}{7 x^{2} - 61 x - 18} = \lim_{x \to 9}\frac{6 x - 45}{14 x - 61} = \frac{6 (9) - 45}{14 (9) - 61} = \frac{9}{65}$