Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{3 x^{3} + 3 x^{2} + 9 x + 5}{5 x^{3} + 4 x^{2} + 7 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{3 x^{3} + 3 x^{2} + 9 x + 5}{5 x^{3} + 4 x^{2} + 7 x - 9} = \lim_{x \to \infty}\frac{9 x^{2} + 6 x + 9}{15 x^{2} + 8 x + 7} = \lim_{x \to \infty}\frac{6 \left(3 x + 1\right)}{2 \left(15 x + 4\right)} = \lim_{x \to \infty}\frac{18}{30} = \frac{3}{5}