Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{- 3 x^{3} - 7 x^{2} + 6 x + 3}{- 2 x^{3} - 8 x^{2} + 4 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{- 3 x^{3} - 7 x^{2} + 6 x + 3}{- 2 x^{3} - 8 x^{2} + 4 x - 6} = \lim_{x \to -\infty}\frac{- 9 x^{2} - 14 x + 6}{- 6 x^{2} - 16 x + 4} = \lim_{x \to -\infty}\frac{- 2 \left(9 x + 7\right)}{- 4 \left(3 x + 4\right)} = \lim_{x \to -\infty}\frac{-18}{-12} = \frac{3}{2}