Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{4 x^{3} - 7 x^{2} + 3 x + 5}{- 9 x^{3} - 2 x^{2} + 5 x + 4}

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} - 7 x^{2} + 3 x + 5}{- 9 x^{3} - 2 x^{2} + 5 x + 4} = \lim_{x \to \infty}\frac{12 x^{2} - 14 x + 3}{- 27 x^{2} - 4 x + 5} = \lim_{x \to \infty}\frac{2 \left(12 x - 7\right)}{- 2 \left(27 x + 2\right)} = \lim_{x \to \infty}\frac{24}{-54} = - \frac{4}{9}