Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{8 x^{3} + 7 x^{2} + 3 x - 8}{- 9 x^{3} + 5 x^{2} - 4 x + 8}

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} + 7 x^{2} + 3 x - 8}{- 9 x^{3} + 5 x^{2} - 4 x + 8} = \lim_{x \to \infty}\frac{24 x^{2} + 14 x + 3}{- 27 x^{2} + 10 x - 4} = \lim_{x \to \infty}\frac{2 \left(24 x + 7\right)}{2 \left(5 - 27 x\right)} = \lim_{x \to \infty}\frac{48}{-54} = - \frac{8}{9}