Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{x^{3} - 5 x^{2} - 9 x + 7}{- 6 x^{3} + 3 x^{2} - 5 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{x^{3} - 5 x^{2} - 9 x + 7}{- 6 x^{3} + 3 x^{2} - 5 x - 2} = \lim_{x \to \infty}\frac{3 x^{2} - 10 x - 9}{- 18 x^{2} + 6 x - 5} = \lim_{x \to \infty}\frac{2 \left(3 x - 5\right)}{6 \left(1 - 6 x\right)} = \lim_{x \to \infty}\frac{6}{-36} = - \frac{1}{6}