Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{6 x^{3} - x^{2} + 7 x - 2}{- 9 x^{3} + 4 x^{2} - 4 x + 6}

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{6 x^{3} - x^{2} + 7 x - 2}{- 9 x^{3} + 4 x^{2} - 4 x + 6} = \lim_{x \to \infty}\frac{18 x^{2} - 2 x + 7}{- 27 x^{2} + 8 x - 4} = \lim_{x \to \infty}\frac{2 \left(18 x - 1\right)}{2 \left(4 - 27 x\right)} = \lim_{x \to \infty}\frac{36}{-54} = - \frac{2}{3}