Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 8 x^{3} + 4 x^{2} - 2 x - 5}{x^{3} + 8 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{- 8 x^{3} + 4 x^{2} - 2 x - 5}{x^{3} + 8 x^{2} - 5 x + 8} = \lim_{x \to \infty}\frac{- 24 x^{2} + 8 x - 2}{3 x^{2} + 16 x - 5} = \lim_{x \to \infty}\frac{8 \left(1 - 6 x\right)}{2 \left(3 x + 8\right)} = \lim_{x \to \infty}\frac{-48}{6} = -8