Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{2 x^{3} - x^{2} - 7 x + 7}{2 x^{3} - 4 x^{2} + 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{2 x^{3} - x^{2} - 7 x + 7}{2 x^{3} - 4 x^{2} + x + 5} = \lim_{x \to \infty}\frac{6 x^{2} - 2 x - 7}{6 x^{2} - 8 x + 1} = \lim_{x \to \infty}\frac{2 \left(6 x - 1\right)}{4 \left(3 x - 2\right)} = \lim_{x \to \infty}\frac{12}{12} = 1