Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{9 x^{3} - 9 x^{2} + 3 x + 9}{- 2 x^{3} - x^{2} - 8 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{9 x^{3} - 9 x^{2} + 3 x + 9}{- 2 x^{3} - x^{2} - 8 x - 8} = \lim_{x \to \infty}\frac{27 x^{2} - 18 x + 3}{- 6 x^{2} - 2 x - 8} = \lim_{x \to \infty}\frac{18 \left(3 x - 1\right)}{- 2 \left(6 x + 1\right)} = \lim_{x \to \infty}\frac{54}{-12} = - \frac{9}{2}