Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{3 x^{3} - 5 x^{2} + 4 x - 7}{7 x^{3} + 4 x^{2} + x + 6}

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{3 x^{3} - 5 x^{2} + 4 x - 7}{7 x^{3} + 4 x^{2} + x + 6} = \lim_{x \to \infty}\frac{9 x^{2} - 10 x + 4}{21 x^{2} + 8 x + 1} = \lim_{x \to \infty}\frac{2 \left(9 x - 5\right)}{2 \left(21 x + 4\right)} = \lim_{x \to \infty}\frac{18}{42} = \frac{3}{7}