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