Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 6 x^{3} + 5 x^{2} - 7 x - 2}{2 x^{3} - 6 x^{2} + 2 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{- 6 x^{3} + 5 x^{2} - 7 x - 2}{2 x^{3} - 6 x^{2} + 2 x + 4} = \lim_{x \to \infty}\frac{- 18 x^{2} + 10 x - 7}{6 x^{2} - 12 x + 2} = \lim_{x \to \infty}\frac{2 \left(5 - 18 x\right)}{12 \left(x - 1\right)} = \lim_{x \to \infty}\frac{-36}{12} = -3