Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 6 x^{3} + 6 x^{2} + 2 x + 1}{- 9 x^{3} + 9 x^{2} - 2 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{- 6 x^{3} + 6 x^{2} + 2 x + 1}{- 9 x^{3} + 9 x^{2} - 2 x + 1} = \lim_{x \to \infty}\frac{- 18 x^{2} + 12 x + 2}{- 27 x^{2} + 18 x - 2} = \lim_{x \to \infty}\frac{12 \left(1 - 3 x\right)}{18 \left(1 - 3 x\right)} = \lim_{x \to \infty}\frac{-36}{-54} = \frac{2}{3}