Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{5 x^{3} - x^{2} + 7 x - 8}{- 8 x^{3} + 7 x^{2} + 3 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{5 x^{3} - x^{2} + 7 x - 8}{- 8 x^{3} + 7 x^{2} + 3 x - 7} = \lim_{x \to -\infty}\frac{15 x^{2} - 2 x + 7}{- 24 x^{2} + 14 x + 3} = \lim_{x \to -\infty}\frac{2 \left(15 x - 1\right)}{2 \left(7 - 24 x\right)} = \lim_{x \to -\infty}\frac{30}{-48} = - \frac{5}{8}