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

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{- 3 x^{3} + 7 x^{2} + 3 x + 9}{3 x^{3} - 9 x^{2} - 9 x + 7} = \lim_{x \to -\infty}\frac{- 9 x^{2} + 14 x + 3}{9 x^{2} - 18 x - 9} = \lim_{x \to -\infty}\frac{2 \left(7 - 9 x\right)}{18 \left(x - 1\right)} = \lim_{x \to -\infty}\frac{-18}{18} = -1