Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 6 x^{3} + 2 x^{2} + x - 9}{- 4 x^{3} + x^{2} + 3 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{- 6 x^{3} + 2 x^{2} + x - 9}{- 4 x^{3} + x^{2} + 3 x - 6} = \lim_{x \to \infty}\frac{- 18 x^{2} + 4 x + 1}{- 12 x^{2} + 2 x + 3} = \lim_{x \to \infty}\frac{4 \left(1 - 9 x\right)}{2 \left(1 - 12 x\right)} = \lim_{x \to \infty}\frac{-36}{-24} = \frac{3}{2}