Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{2 x^{3} - 6 x^{2} + 9 x + 6}{- 7 x^{3} - 2 x^{2} - 3 x + 3}

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} - 6 x^{2} + 9 x + 6}{- 7 x^{3} - 2 x^{2} - 3 x + 3} = \lim_{x \to -\infty}\frac{6 x^{2} - 12 x + 9}{- 21 x^{2} - 4 x - 3} = \lim_{x \to -\infty}\frac{12 \left(x - 1\right)}{- 2 \left(21 x + 2\right)} = \lim_{x \to -\infty}\frac{12}{-42} = - \frac{2}{7}