Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{9 x^{3} - 7 x^{2} - 8 x - 2}{2 x^{3} - 7 x^{2} - x + 2}

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{9 x^{3} - 7 x^{2} - 8 x - 2}{2 x^{3} - 7 x^{2} - x + 2} = \lim_{x \to -\infty}\frac{27 x^{2} - 14 x - 8}{6 x^{2} - 14 x - 1} = \lim_{x \to -\infty}\frac{2 \left(27 x - 7\right)}{2 \left(6 x - 7\right)} = \lim_{x \to -\infty}\frac{54}{12} = \frac{9}{2}