Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{8 x^{3} + 4 x^{2} + 4 x - 5}{- 5 x^{3} + 2 x^{2} + 9 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{8 x^{3} + 4 x^{2} + 4 x - 5}{- 5 x^{3} + 2 x^{2} + 9 x + 6} = \lim_{x \to -\infty}\frac{24 x^{2} + 8 x + 4}{- 15 x^{2} + 4 x + 9} = \lim_{x \to -\infty}\frac{8 \left(6 x + 1\right)}{2 \left(2 - 15 x\right)} = \lim_{x \to -\infty}\frac{48}{-30} = - \frac{8}{5}