Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{- 6 x^{3} - 8 x^{2} - 3 x - 6}{- 4 x^{3} + 7 x^{2} + 8 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{- 6 x^{3} - 8 x^{2} - 3 x - 6}{- 4 x^{3} + 7 x^{2} + 8 x - 9} = \lim_{x \to \infty}\frac{- 18 x^{2} - 16 x - 3}{- 12 x^{2} + 14 x + 8} = \lim_{x \to \infty}\frac{- 4 \left(9 x + 4\right)}{2 \left(7 - 12 x\right)} = \lim_{x \to \infty}\frac{-36}{-24} = \frac{3}{2}