Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -1}\frac{2 x^{2} + 14 x + 12}{3 x^{2} - x - 4}$

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: $LaTeX: \lim_{x \to -1}\frac{2 x^{2} + 14 x + 12}{3 x^{2} - x - 4} = \lim_{x \to -1}\frac{4 x + 14}{6 x - 1} = \frac{4 (-1) + 14}{6 (-1) - 1} = - \frac{10}{7}$