Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{8 x^{3} + x^{2} + 9 x - 6}{2 x^{3} - 2 x^{2} - 9 x - 7}

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{8 x^{3} + x^{2} + 9 x - 6}{2 x^{3} - 2 x^{2} - 9 x - 7} = \lim_{x \to -\infty}\frac{24 x^{2} + 2 x + 9}{6 x^{2} - 4 x - 9} = \lim_{x \to -\infty}\frac{2 \left(24 x + 1\right)}{4 \left(3 x - 1\right)} = \lim_{x \to -\infty}\frac{48}{12} = 4