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

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 - 7}{9 x^{3} + 3 x^{2} - 9 x - 8} = \lim_{x \to \infty}\frac{- 18 x^{2} - 16 x + 5}{27 x^{2} + 6 x - 9} = \lim_{x \to \infty}\frac{- 4 \left(9 x + 4\right)}{6 \left(9 x + 1\right)} = \lim_{x \to \infty}\frac{-36}{54} = - \frac{2}{3}