Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{3 x^{3} + 2 x^{2} + 6 x + 8}{5 x^{3} + 6 x^{2} + 3 x + 7}

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} + 2 x^{2} + 6 x + 8}{5 x^{3} + 6 x^{2} + 3 x + 7} = \lim_{x \to \infty}\frac{9 x^{2} + 4 x + 6}{15 x^{2} + 12 x + 3} = \lim_{x \to \infty}\frac{2 \left(9 x + 2\right)}{6 \left(5 x + 2\right)} = \lim_{x \to \infty}\frac{18}{30} = \frac{3}{5}