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

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{- 4 x^{3} + x^{2} + 4 x - 1}{2 x^{3} + 9 x^{2} + 7 x + 4} = \lim_{x \to \infty}\frac{- 12 x^{2} + 2 x + 4}{6 x^{2} + 18 x + 7} = \lim_{x \to \infty}\frac{2 \left(1 - 12 x\right)}{6 \left(2 x + 3\right)} = \lim_{x \to \infty}\frac{-24}{12} = -2