Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- x^{3} + 7 x^{2} - 8 x - 9}{- 3 x^{3} + 4 x^{2} - 7 x + 1}

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{- x^{3} + 7 x^{2} - 8 x - 9}{- 3 x^{3} + 4 x^{2} - 7 x + 1} = \lim_{x \to \infty}\frac{- 3 x^{2} + 14 x - 8}{- 9 x^{2} + 8 x - 7} = \lim_{x \to \infty}\frac{2 \left(7 - 3 x\right)}{2 \left(4 - 9 x\right)} = \lim_{x \to \infty}\frac{-6}{-18} = \frac{1}{3}