Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{- 3 x^{2} + x + 2}{- 9 x^{2} + 4 x + 8}

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{- 3 x^{2} + x + 2}{- 9 x^{2} + 4 x + 8} = \lim_{x \to -\infty}\frac{1 - 6 x}{4 - 18 x} = \lim_{x \to -\infty}\frac{-6}{-18} = \frac{1}{3}