Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{2 x^{3} - 3 x^{2} - 2 x - 2}{8 x^{3} - 4 x^{2} - 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{2 x^{3} - 3 x^{2} - 2 x - 2}{8 x^{3} - 4 x^{2} - x - 9} = \lim_{x \to -\infty}\frac{6 x^{2} - 6 x - 2}{24 x^{2} - 8 x - 1} = \lim_{x \to -\infty}\frac{6 \left(2 x - 1\right)}{8 \left(6 x - 1\right)} = \lim_{x \to -\infty}\frac{12}{48} = \frac{1}{4}