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

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} + 7 x^{2} - 3 x - 2}{3 x^{3} + 5 x^{2} - 5 x + 7} = \lim_{x \to \infty}\frac{- 24 x^{2} + 14 x - 3}{9 x^{2} + 10 x - 5} = \lim_{x \to \infty}\frac{2 \left(7 - 24 x\right)}{2 \left(9 x + 5\right)} = \lim_{x \to \infty}\frac{-48}{18} = - \frac{8}{3}