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

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{5 x^{3} - x^{2} + 3 x + 3}{7 x^{3} - x^{2} - 6 x + 2} = \lim_{x \to \infty}\frac{15 x^{2} - 2 x + 3}{21 x^{2} - 2 x - 6} = \lim_{x \to \infty}\frac{2 \left(15 x - 1\right)}{2 \left(21 x - 1\right)} = \lim_{x \to \infty}\frac{30}{42} = \frac{5}{7}