Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 5 x^{3} + 9 x^{2} - 6 x - 9}{9 x^{3} - 8 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{- 5 x^{3} + 9 x^{2} - 6 x - 9}{9 x^{3} - 8 x^{2} + 3 x - 9} = \lim_{x \to \infty}\frac{- 15 x^{2} + 18 x - 6}{27 x^{2} - 16 x + 3} = \lim_{x \to \infty}\frac{6 \left(3 - 5 x\right)}{2 \left(27 x - 8\right)} = \lim_{x \to \infty}\frac{-30}{54} = - \frac{5}{9}