Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 6 x^{3} + 8 x^{2} + 5 x - 5}{3 x^{3} + 7 x^{2} - 3 x - 9}

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{- 6 x^{3} + 8 x^{2} + 5 x - 5}{3 x^{3} + 7 x^{2} - 3 x - 9} = \lim_{x \to \infty}\frac{- 18 x^{2} + 16 x + 5}{9 x^{2} + 14 x - 3} = \lim_{x \to \infty}\frac{4 \left(4 - 9 x\right)}{2 \left(9 x + 7\right)} = \lim_{x \to \infty}\frac{-36}{18} = -2