Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{- x^{3} + 9 x^{2} + 5 x + 9}{- 7 x^{3} + 5 x^{2} - 4 x + 9}

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{- x^{3} + 9 x^{2} + 5 x + 9}{- 7 x^{3} + 5 x^{2} - 4 x + 9} = \lim_{x \to -\infty}\frac{- 3 x^{2} + 18 x + 5}{- 21 x^{2} + 10 x - 4} = \lim_{x \to -\infty}\frac{6 \left(3 - x\right)}{2 \left(5 - 21 x\right)} = \lim_{x \to -\infty}\frac{-6}{-42} = \frac{1}{7}