Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 3}\frac{3 x^{2} - 27}{- x^{2} + 9 x - 18}$

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: $LaTeX: \lim_{x \to 3}\frac{3 x^{2} - 27}{- x^{2} + 9 x - 18} = \lim_{x \to 3}\frac{6 x}{9 - 2 x} = \frac{6 (3)}{9 - 2 (3)} = 6$