Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{7 x^{2} + 8 x + 7}{4 x^{2} - x - 9}

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{7 x^{2} + 8 x + 7}{4 x^{2} - x - 9} = \lim_{x \to \infty}\frac{14 x + 8}{8 x - 1} = \lim_{x \to \infty}\frac{14}{8} = \frac{7}{4}