Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{4 x^{3} + 3 x^{2} - 3 x - 8}{- 3 x^{3} + 7 x^{2} - 6 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{4 x^{3} + 3 x^{2} - 3 x - 8}{- 3 x^{3} + 7 x^{2} - 6 x + 7} = \lim_{x \to \infty}\frac{12 x^{2} + 6 x - 3}{- 9 x^{2} + 14 x - 6} = \lim_{x \to \infty}\frac{6 \left(4 x + 1\right)}{2 \left(7 - 9 x\right)} = \lim_{x \to \infty}\frac{24}{-18} = - \frac{4}{3}