Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 1}\frac{2 x^{2} - 6 x + 4}{7 x^{2} + 2 x - 9}$

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: $LaTeX: \lim_{x \to 1}\frac{2 x^{2} - 6 x + 4}{7 x^{2} + 2 x - 9} = \lim_{x \to 1}\frac{4 x - 6}{14 x + 2} = \frac{4 (1) - 6}{14 (1) + 2} = - \frac{1}{8}$