Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{9 x^{3} + 5 x^{2} + 9 x - 9}{- 5 x^{3} + x^{2} - 7 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{9 x^{3} + 5 x^{2} + 9 x - 9}{- 5 x^{3} + x^{2} - 7 x + 3} = \lim_{x \to \infty}\frac{27 x^{2} + 10 x + 9}{- 15 x^{2} + 2 x - 7} = \lim_{x \to \infty}\frac{2 \left(27 x + 5\right)}{2 \left(1 - 15 x\right)} = \lim_{x \to \infty}\frac{54}{-30} = - \frac{9}{5}