Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{3 x^{2} + 6 x + 9}{- 5 x^{2} - 2 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{3 x^{2} + 6 x + 9}{- 5 x^{2} - 2 x - 4} = \lim_{x \to \infty}\frac{6 x + 6}{- 10 x - 2} = \lim_{x \to \infty}\frac{6}{-10} = - \frac{3}{5}