Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{x^{2} - 2 x + 4}{- 3 x^{2} - 5 x + 7}

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{x^{2} - 2 x + 4}{- 3 x^{2} - 5 x + 7} = \lim_{x \to \infty}\frac{2 x - 2}{- 6 x - 5} = \lim_{x \to \infty}\frac{2}{-6} = - \frac{1}{3}