Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 1}\frac{2 x^{2} - 14 x + 12}{- x^{2} + 9 x - 8}$

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: $LaTeX: \lim_{x \to 1}\frac{2 x^{2} - 14 x + 12}{- x^{2} + 9 x - 8} = \lim_{x \to 1}\frac{4 x - 14}{9 - 2 x} = \frac{4 (1) - 14}{9 - 2 (1)} = - \frac{10}{7}$