Evaluate the limit $LaTeX: \displaystyle \lim_{x \to -9}\frac{5 x^{2} + 35 x - 90}{6 x^{2} + 55 x + 9}$

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: $LaTeX: \lim_{x \to -9}\frac{5 x^{2} + 35 x - 90}{6 x^{2} + 55 x + 9} = \lim_{x \to -9}\frac{10 x + 35}{12 x + 55} = \frac{10 (-9) + 35}{12 (-9) + 55} = \frac{55}{53}$