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

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{x^{3} - 8 x^{2} - 7 x + 2}{- 7 x^{3} - 2 x^{2} - 7 x + 7} = \lim_{x \to -\infty}\frac{3 x^{2} - 16 x - 7}{- 21 x^{2} - 4 x - 7} = \lim_{x \to -\infty}\frac{2 \left(3 x - 8\right)}{- 2 \left(21 x + 2\right)} = \lim_{x \to -\infty}\frac{6}{-42} = - \frac{1}{7}