Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{5 x^{3} + 3 x^{2} - 8 x + 4}{- 2 x^{3} + 7 x^{2} + 4 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} + 3 x^{2} - 8 x + 4}{- 2 x^{3} + 7 x^{2} + 4 x + 7} = \lim_{x \to \infty}\frac{15 x^{2} + 6 x - 8}{- 6 x^{2} + 14 x + 4} = \lim_{x \to \infty}\frac{6 \left(5 x + 1\right)}{2 \left(7 - 6 x\right)} = \lim_{x \to \infty}\frac{30}{-12} = - \frac{5}{2}