Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 6}\frac{5 x^{2} - 65 x + 210}{- 3 x^{2} + 21 x - 18}$

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: $LaTeX: \lim_{x \to 6}\frac{5 x^{2} - 65 x + 210}{- 3 x^{2} + 21 x - 18} = \lim_{x \to 6}\frac{10 x - 65}{21 - 6 x} = \frac{10 (6) - 65}{21 - 6 (6)} = \frac{1}{3}$