Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{5 x^{3} + 8 x^{2} + x - 3}{- 8 x^{3} + 6 x^{2} + 9 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{5 x^{3} + 8 x^{2} + x - 3}{- 8 x^{3} + 6 x^{2} + 9 x - 3} = \lim_{x \to -\infty}\frac{15 x^{2} + 16 x + 1}{- 24 x^{2} + 12 x + 9} = \lim_{x \to -\infty}\frac{2 \left(15 x + 8\right)}{12 \left(1 - 4 x\right)} = \lim_{x \to -\infty}\frac{30}{-48} = - \frac{5}{8}