Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 4 x^{2} + 2 x + 3}{- 9 x^{2} + x - 3}

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{- 4 x^{2} + 2 x + 3}{- 9 x^{2} + x - 3} = \lim_{x \to \infty}\frac{2 - 8 x}{1 - 18 x} = \lim_{x \to \infty}\frac{-8}{-18} = \frac{4}{9}