Evaluate the limit $LaTeX: \displaystyle \lim_{x \to 4}\frac{2 x^{2} - 2 x - 24}{x^{2} - 3 x - 4}$

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: $LaTeX: \lim_{x \to 4}\frac{2 x^{2} - 2 x - 24}{x^{2} - 3 x - 4} = \lim_{x \to 4}\frac{4 x - 2}{2 x - 3} = \frac{4 (4) - 2}{2 (4) - 3} = \frac{14}{5}$