Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{7 x^{3} - 4 x^{2} - 6 x - 2}{- 6 x^{3} - 5 x^{2} + 3 x + 3}

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{7 x^{3} - 4 x^{2} - 6 x - 2}{- 6 x^{3} - 5 x^{2} + 3 x + 3} = \lim_{x \to \infty}\frac{21 x^{2} - 8 x - 6}{- 18 x^{2} - 10 x + 3} = \lim_{x \to \infty}\frac{2 \left(21 x - 4\right)}{- 2 \left(18 x + 5\right)} = \lim_{x \to \infty}\frac{42}{-36} = - \frac{7}{6}