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

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} + 7 x^{2} + 2 x - 7}{2 x^{3} + 5 x^{2} + 7 x + 5} = \lim_{x \to -\infty}\frac{- 15 x^{2} + 14 x + 2}{6 x^{2} + 10 x + 7} = \lim_{x \to -\infty}\frac{2 \left(7 - 15 x\right)}{2 \left(6 x + 5\right)} = \lim_{x \to -\infty}\frac{-30}{12} = - \frac{5}{2}