Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{- 7 x^{3} - 3 x^{2} + 4 x + 9}{5 x^{3} + 5 x^{2} + 9 x + 6}

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} - 3 x^{2} + 4 x + 9}{5 x^{3} + 5 x^{2} + 9 x + 6} = \lim_{x \to -\infty}\frac{- 21 x^{2} - 6 x + 4}{15 x^{2} + 10 x + 9} = \lim_{x \to -\infty}\frac{- 6 \left(7 x + 1\right)}{10 \left(3 x + 1\right)} = \lim_{x \to -\infty}\frac{-42}{30} = - \frac{7}{5}