Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{9 x^{3} + 7 x^{2} + 9 x + 6}{- 4 x^{3} + x^{2} + 3 x + 9}

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{9 x^{3} + 7 x^{2} + 9 x + 6}{- 4 x^{3} + x^{2} + 3 x + 9} = \lim_{x \to \infty}\frac{27 x^{2} + 14 x + 9}{- 12 x^{2} + 2 x + 3} = \lim_{x \to \infty}\frac{2 \left(27 x + 7\right)}{2 \left(1 - 12 x\right)} = \lim_{x \to \infty}\frac{54}{-24} = - \frac{9}{4}