Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{8 x^{3} - 9 x^{2} + 5 x - 6}{7 x^{3} + 3 x^{2} + 6 x - 2}

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} - 9 x^{2} + 5 x - 6}{7 x^{3} + 3 x^{2} + 6 x - 2} = \lim_{x \to \infty}\frac{24 x^{2} - 18 x + 5}{21 x^{2} + 6 x + 6} = \lim_{x \to \infty}\frac{6 \left(8 x - 3\right)}{6 \left(7 x + 1\right)} = \lim_{x \to \infty}\frac{48}{42} = \frac{8}{7}