Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 1}\frac{3 x^{2} - 12 x + 9}{- 2 x^{2} + x + 1}$

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: $LaTeX: \lim_{x \to 1}\frac{3 x^{2} - 12 x + 9}{- 2 x^{2} + x + 1} = \lim_{x \to 1}\frac{6 x - 12}{1 - 4 x} = \frac{6 (1) - 12}{1 - 4 (1)} = 2$