Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{4 x^{3} - 3 x^{2} + 8 x - 1}{- 9 x^{3} + 6 x^{2} - 9 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} - 3 x^{2} + 8 x - 1}{- 9 x^{3} + 6 x^{2} - 9 x + 4} = \lim_{x \to \infty}\frac{12 x^{2} - 6 x + 8}{- 27 x^{2} + 12 x - 9} = \lim_{x \to \infty}\frac{6 \left(4 x - 1\right)}{6 \left(2 - 9 x\right)} = \lim_{x \to \infty}\frac{24}{-54} = - \frac{4}{9}