Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 5 x^{2} + 9 x + 2}{9 x^{2} + 4 x + 7}

The limit is an indeterminate form of the type LaTeX: \displaystyle \frac{\infty}{\infty} . Using L'Hospitial's rule 2 times gives: LaTeX: \lim_{x \to \infty}\frac{- 5 x^{2} + 9 x + 2}{9 x^{2} + 4 x + 7} = \lim_{x \to \infty}\frac{9 - 10 x}{18 x + 4} = \lim_{x \to \infty}\frac{-10}{18} = - \frac{5}{9}