Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -1}\frac{3 x^{2} - 3}{4 x^{2} - 5 x - 9}$

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: $LaTeX: \lim_{x \to -1}\frac{3 x^{2} - 3}{4 x^{2} - 5 x - 9} = \lim_{x \to -1}\frac{6 x}{8 x - 5} = \frac{6 (-1)}{8 (-1) - 5} = \frac{6}{13}$