Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{3 x^{3} + 8 x^{2} + 5 x - 3}{8 x^{3} - 6 x^{2} + 5 x - 8}

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{3 x^{3} + 8 x^{2} + 5 x - 3}{8 x^{3} - 6 x^{2} + 5 x - 8} = \lim_{x \to -\infty}\frac{9 x^{2} + 16 x + 5}{24 x^{2} - 12 x + 5} = \lim_{x \to -\infty}\frac{2 \left(9 x + 8\right)}{12 \left(4 x - 1\right)} = \lim_{x \to -\infty}\frac{18}{48} = \frac{3}{8}