Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 8 x^{3} - 4 x^{2} + x + 8}{2 x^{3} - 5 x^{2} - x - 3}

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} + x + 8}{2 x^{3} - 5 x^{2} - x - 3} = \lim_{x \to \infty}\frac{- 24 x^{2} - 8 x + 1}{6 x^{2} - 10 x - 1} = \lim_{x \to \infty}\frac{- 8 \left(6 x + 1\right)}{2 \left(6 x - 5\right)} = \lim_{x \to \infty}\frac{-48}{12} = -4