Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 9 x^{3} - 6 x^{2} - 5 x - 4}{9 x^{3} + x^{2} - 6 x - 6}

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{- 9 x^{3} - 6 x^{2} - 5 x - 4}{9 x^{3} + x^{2} - 6 x - 6} = \lim_{x \to \infty}\frac{- 27 x^{2} - 12 x - 5}{27 x^{2} + 2 x - 6} = \lim_{x \to \infty}\frac{- 6 \left(9 x + 2\right)}{2 \left(27 x + 1\right)} = \lim_{x \to \infty}\frac{-54}{54} = -1