Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -1}\frac{5 x^{2} + 40 x + 35}{- x^{2} + 8 x + 9}$

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: $LaTeX: \lim_{x \to -1}\frac{5 x^{2} + 40 x + 35}{- x^{2} + 8 x + 9} = \lim_{x \to -1}\frac{10 x + 40}{8 - 2 x} = \frac{10 (-1) + 40}{8 - 2 (-1)} = 3$