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

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{- 2 x^{3} + 5 x^{2} + 5 x - 8}{- 5 x^{3} - 5 x^{2} + 5 x - 8} = \lim_{x \to -\infty}\frac{- 6 x^{2} + 10 x + 5}{- 15 x^{2} - 10 x + 5} = \lim_{x \to -\infty}\frac{2 \left(5 - 6 x\right)}{- 10 \left(3 x + 1\right)} = \lim_{x \to -\infty}\frac{-12}{-30} = \frac{2}{5}