Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 4 x^{3} + 8 x^{2} - 5 x - 7}{- 7 x^{3} + x^{2} - 7 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{- 4 x^{3} + 8 x^{2} - 5 x - 7}{- 7 x^{3} + x^{2} - 7 x - 8} = \lim_{x \to \infty}\frac{- 12 x^{2} + 16 x - 5}{- 21 x^{2} + 2 x - 7} = \lim_{x \to \infty}\frac{8 \left(2 - 3 x\right)}{2 \left(1 - 21 x\right)} = \lim_{x \to \infty}\frac{-24}{-42} = \frac{4}{7}