Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{5 x^{3} + 7 x^{2} + 4 x + 8}{- x^{3} - 2 x^{2} - 9 x + 4}

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} + 7 x^{2} + 4 x + 8}{- x^{3} - 2 x^{2} - 9 x + 4} = \lim_{x \to -\infty}\frac{15 x^{2} + 14 x + 4}{- 3 x^{2} - 4 x - 9} = \lim_{x \to -\infty}\frac{2 \left(15 x + 7\right)}{- 2 \left(3 x + 2\right)} = \lim_{x \to -\infty}\frac{30}{-6} = -5