Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 9 x^{3} - 3 x^{2} + x + 5}{6 x^{3} + 6 x^{2} + 4 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} - 3 x^{2} + x + 5}{6 x^{3} + 6 x^{2} + 4 x - 6} = \lim_{x \to \infty}\frac{- 27 x^{2} - 6 x + 1}{18 x^{2} + 12 x + 4} = \lim_{x \to \infty}\frac{- 6 \left(9 x + 1\right)}{12 \left(3 x + 1\right)} = \lim_{x \to \infty}\frac{-54}{36} = - \frac{3}{2}