Evaluate the limit LaTeX: \displaystyle \lim_{x \to -\infty}\frac{x^{3} - x^{2} + 7 x - 5}{7 x^{3} - 3 x^{2} - 9 x - 2}

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{x^{3} - x^{2} + 7 x - 5}{7 x^{3} - 3 x^{2} - 9 x - 2} = \lim_{x \to -\infty}\frac{3 x^{2} - 2 x + 7}{21 x^{2} - 6 x - 9} = \lim_{x \to -\infty}\frac{2 \left(3 x - 1\right)}{6 \left(7 x - 1\right)} = \lim_{x \to -\infty}\frac{6}{42} = \frac{1}{7}