Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -1}\frac{3 x^{2} - 6 x - 9}{- 8 x^{2} + x + 9}$

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: $LaTeX: \lim_{x \to -1}\frac{3 x^{2} - 6 x - 9}{- 8 x^{2} + x + 9} = \lim_{x \to -1}\frac{6 x - 6}{1 - 16 x} = \frac{6 (-1) - 6}{1 - 16 (-1)} = - \frac{12}{17}$