Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 2 x^{3} + 9 x^{2} + 7 x + 4}{3 x^{3} - 7 x^{2} + 6 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{- 2 x^{3} + 9 x^{2} + 7 x + 4}{3 x^{3} - 7 x^{2} + 6 x + 8} = \lim_{x \to \infty}\frac{- 6 x^{2} + 18 x + 7}{9 x^{2} - 14 x + 6} = \lim_{x \to \infty}\frac{6 \left(3 - 2 x\right)}{2 \left(9 x - 7\right)} = \lim_{x \to \infty}\frac{-12}{18} = - \frac{2}{3}