Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{8 x^{3} - 8 x^{2} + 2 x - 9}{- 5 x^{3} - 6 x^{2} - 7 x + 2}

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} - 8 x^{2} + 2 x - 9}{- 5 x^{3} - 6 x^{2} - 7 x + 2} = \lim_{x \to \infty}\frac{24 x^{2} - 16 x + 2}{- 15 x^{2} - 12 x - 7} = \lim_{x \to \infty}\frac{16 \left(3 x - 1\right)}{- 6 \left(5 x + 2\right)} = \lim_{x \to \infty}\frac{48}{-30} = - \frac{8}{5}