Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -1}\frac{5 x^{2} - 20 x - 25}{- 9 x^{2} - x + 8}$

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: $LaTeX: \lim_{x \to -1}\frac{5 x^{2} - 20 x - 25}{- 9 x^{2} - x + 8} = \lim_{x \to -1}\frac{10 x - 20}{- 18 x - 1} = \frac{10 (-1) - 20}{- 18 (-1) - 1} = - \frac{30}{17}$