Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -1}\frac{5 x^{2} + 25 x + 20}{- 9 x^{2} - 3 x + 6}$

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: $LaTeX: \lim_{x \to -1}\frac{5 x^{2} + 25 x + 20}{- 9 x^{2} - 3 x + 6} = \lim_{x \to -1}\frac{10 x + 25}{- 18 x - 3} = \frac{10 (-1) + 25}{- 18 (-1) - 3} = 1$