Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -1}\frac{2 x^{2} - 2 x - 4}{- x^{2} - 10 x - 9}$

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: $LaTeX: \lim_{x \to -1}\frac{2 x^{2} - 2 x - 4}{- x^{2} - 10 x - 9} = \lim_{x \to -1}\frac{4 x - 2}{- 2 x - 10} = \frac{4 (-1) - 2}{- 2 (-1) - 10} = \frac{3}{4}$