Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{3 x^{3} + 7 x^{2} + 3 x - 9}{3 x^{3} + 9 x^{2} - 6 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{3 x^{3} + 7 x^{2} + 3 x - 9}{3 x^{3} + 9 x^{2} - 6 x + 4} = \lim_{x \to \infty}\frac{9 x^{2} + 14 x + 3}{9 x^{2} + 18 x - 6} = \lim_{x \to \infty}\frac{2 \left(9 x + 7\right)}{18 \left(x + 1\right)} = \lim_{x \to \infty}\frac{18}{18} = 1