Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 3}\frac{3 x^{2} - 30 x + 63}{- 2 x^{2} + 7 x - 3}$

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: $LaTeX: \lim_{x \to 3}\frac{3 x^{2} - 30 x + 63}{- 2 x^{2} + 7 x - 3} = \lim_{x \to 3}\frac{6 x - 30}{7 - 4 x} = \frac{6 (3) - 30}{7 - 4 (3)} = \frac{12}{5}$