Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 4 x^{3} - 5 x^{2} + 6 x - 6}{- 6 x^{3} - x^{2} + x + 4}

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{- 4 x^{3} - 5 x^{2} + 6 x - 6}{- 6 x^{3} - x^{2} + x + 4} = \lim_{x \to \infty}\frac{- 12 x^{2} - 10 x + 6}{- 18 x^{2} - 2 x + 1} = \lim_{x \to \infty}\frac{- 2 \left(12 x + 5\right)}{- 2 \left(18 x + 1\right)} = \lim_{x \to \infty}\frac{-24}{-36} = \frac{2}{3}