Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{- x^{3} - 2 x^{2} + 2 x + 8}{- 5 x^{3} - 9 x^{2} - 9 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{- x^{3} - 2 x^{2} + 2 x + 8}{- 5 x^{3} - 9 x^{2} - 9 x - 6} = \lim_{x \to -\infty}\frac{- 3 x^{2} - 4 x + 2}{- 15 x^{2} - 18 x - 9} = \lim_{x \to -\infty}\frac{- 2 \left(3 x + 2\right)}{- 6 \left(5 x + 3\right)} = \lim_{x \to -\infty}\frac{-6}{-30} = \frac{1}{5}