Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -1}\frac{2 x^{2} - 6 x - 8}{- 3 x^{2} - 10 x - 7}$

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: $LaTeX: \lim_{x \to -1}\frac{2 x^{2} - 6 x - 8}{- 3 x^{2} - 10 x - 7} = \lim_{x \to -1}\frac{4 x - 6}{- 6 x - 10} = \frac{4 (-1) - 6}{- 6 (-1) - 10} = \frac{5}{2}$