Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{x^{3} - 4 x^{2} - 3 x + 5}{7 x^{3} - 7 x^{2} + 9 x + 1}

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} - 4 x^{2} - 3 x + 5}{7 x^{3} - 7 x^{2} + 9 x + 1} = \lim_{x \to \infty}\frac{3 x^{2} - 8 x - 3}{21 x^{2} - 14 x + 9} = \lim_{x \to \infty}\frac{2 \left(3 x - 4\right)}{14 \left(3 x - 1\right)} = \lim_{x \to \infty}\frac{6}{42} = \frac{1}{7}