Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{- 3 x^{3} - 2 x^{2} + 7 x + 8}{9 x^{3} + 5 x^{2} + 2 x + 6}

The limit is an indeterminate form of the type LaTeX: \displaystyle \frac{\infty}{\infty} . Using L'Hospitial's rule 3 times gives: LaTeX: \lim_{x \to -\infty}\frac{- 3 x^{3} - 2 x^{2} + 7 x + 8}{9 x^{3} + 5 x^{2} + 2 x + 6} = \lim_{x \to -\infty}\frac{- 9 x^{2} - 4 x + 7}{27 x^{2} + 10 x + 2} = \lim_{x \to -\infty}\frac{- 2 \left(9 x + 2\right)}{2 \left(27 x + 5\right)} = \lim_{x \to -\infty}\frac{-18}{54} = - \frac{1}{3}