Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{9 x^{3} + 6 x^{2} + 4 x - 2}{3 x^{3} - 3 x^{2} + 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{9 x^{3} + 6 x^{2} + 4 x - 2}{3 x^{3} - 3 x^{2} + x + 8} = \lim_{x \to -\infty}\frac{27 x^{2} + 12 x + 4}{9 x^{2} - 6 x + 1} = \lim_{x \to -\infty}\frac{6 \left(9 x + 2\right)}{6 \left(3 x - 1\right)} = \lim_{x \to -\infty}\frac{54}{18} = 3