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

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