Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 7 x^{3} + 5 x^{2} - 7 x - 6}{- 3 x^{3} + 5 x^{2} + 2 x - 8}

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} + 5 x^{2} - 7 x - 6}{- 3 x^{3} + 5 x^{2} + 2 x - 8} = \lim_{x \to \infty}\frac{- 21 x^{2} + 10 x - 7}{- 9 x^{2} + 10 x + 2} = \lim_{x \to \infty}\frac{2 \left(5 - 21 x\right)}{2 \left(5 - 9 x\right)} = \lim_{x \to \infty}\frac{-42}{-18} = \frac{7}{3}