Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{7 x^{3} - 8 x^{2} - 6 x + 1}{7 x^{3} + 7 x^{2} + 5 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{7 x^{3} - 8 x^{2} - 6 x + 1}{7 x^{3} + 7 x^{2} + 5 x - 1} = \lim_{x \to \infty}\frac{21 x^{2} - 16 x - 6}{21 x^{2} + 14 x + 5} = \lim_{x \to \infty}\frac{2 \left(21 x - 8\right)}{14 \left(3 x + 1\right)} = \lim_{x \to \infty}\frac{42}{42} = 1