Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{4 x^{2} + 3 x - 1}{x^{2} + 6 x + 4}

The limit is an indeterminate form of the type LaTeX: \displaystyle \frac{\infty}{\infty} . Using L'Hospitial's rule 2 times gives: LaTeX: \lim_{x \to \infty}\frac{4 x^{2} + 3 x - 1}{x^{2} + 6 x + 4} = \lim_{x \to \infty}\frac{8 x + 3}{2 x + 6} = \lim_{x \to \infty}\frac{8}{2} = 4