Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{- 6 x^{3} - 8 x^{2} - 4 x + 8}{4 x^{3} - 2 x^{2} + 9 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{- 6 x^{3} - 8 x^{2} - 4 x + 8}{4 x^{3} - 2 x^{2} + 9 x - 8} = \lim_{x \to -\infty}\frac{- 18 x^{2} - 16 x - 4}{12 x^{2} - 4 x + 9} = \lim_{x \to -\infty}\frac{- 4 \left(9 x + 4\right)}{4 \left(6 x - 1\right)} = \lim_{x \to -\infty}\frac{-36}{24} = - \frac{3}{2}