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