Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{5 x^{3} + 5 x^{2} - 7 x - 2}{3 x^{3} + 5 x^{2} - 8 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} + 5 x^{2} - 7 x - 2}{3 x^{3} + 5 x^{2} - 8 x - 2} = \lim_{x \to \infty}\frac{15 x^{2} + 10 x - 7}{9 x^{2} + 10 x - 8} = \lim_{x \to \infty}\frac{10 \left(3 x + 1\right)}{2 \left(9 x + 5\right)} = \lim_{x \to \infty}\frac{30}{18} = \frac{5}{3}