Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{4 x^{3} - 2 x^{2} + 8 x + 7}{- 3 x^{3} + 9 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{4 x^{3} - 2 x^{2} + 8 x + 7}{- 3 x^{3} + 9 x^{2} - 3 x - 7} = \lim_{x \to -\infty}\frac{12 x^{2} - 4 x + 8}{- 9 x^{2} + 18 x - 3} = \lim_{x \to -\infty}\frac{4 \left(6 x - 1\right)}{18 \left(1 - x\right)} = \lim_{x \to -\infty}\frac{24}{-18} = - \frac{4}{3}