Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 7 x^{3} - 7 x^{2} - 9 x + 2}{- 7 x^{3} + 8 x^{2} + 5 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} - 7 x^{2} - 9 x + 2}{- 7 x^{3} + 8 x^{2} + 5 x + 8} = \lim_{x \to \infty}\frac{- 21 x^{2} - 14 x - 9}{- 21 x^{2} + 16 x + 5} = \lim_{x \to \infty}\frac{- 14 \left(3 x + 1\right)}{2 \left(8 - 21 x\right)} = \lim_{x \to \infty}\frac{-42}{-42} = 1