Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4}$

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: $LaTeX: \lim_{x \to -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4} = \lim_{x \to -2}\frac{4 x - 6}{- 8 x - 6} = \frac{4 (-2) - 6}{- 8 (-2) - 6} = - \frac{7}{5}$