Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{x^{3} + 9 x^{2} - 6 x - 3}{9 x^{3} - 5 x^{2} - 9 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{x^{3} + 9 x^{2} - 6 x - 3}{9 x^{3} - 5 x^{2} - 9 x + 3} = \lim_{x \to -\infty}\frac{3 x^{2} + 18 x - 6}{27 x^{2} - 10 x - 9} = \lim_{x \to -\infty}\frac{6 \left(x + 3\right)}{2 \left(27 x - 5\right)} = \lim_{x \to -\infty}\frac{6}{54} = \frac{1}{9}