Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 2}\frac{2 x^{2} + 4 x - 16}{5 x^{2} - 13 x + 6}$

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: $LaTeX: \lim_{x \to 2}\frac{2 x^{2} + 4 x - 16}{5 x^{2} - 13 x + 6} = \lim_{x \to 2}\frac{4 x + 4}{10 x - 13} = \frac{4 (2) + 4}{10 (2) - 13} = \frac{12}{7}$