Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{9 x^{3} - 9 x^{2} + 4 x - 6}{- 8 x^{3} + 2 x^{2} - 5 x - 1}

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} + 4 x - 6}{- 8 x^{3} + 2 x^{2} - 5 x - 1} = \lim_{x \to \infty}\frac{27 x^{2} - 18 x + 4}{- 24 x^{2} + 4 x - 5} = \lim_{x \to \infty}\frac{18 \left(3 x - 1\right)}{4 \left(1 - 12 x\right)} = \lim_{x \to \infty}\frac{54}{-48} = - \frac{9}{8}