Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{2 x^{3} + 2 x^{2} - 7 x - 4}{2 x^{3} + 8 x^{2} + 6 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{2 x^{3} + 2 x^{2} - 7 x - 4}{2 x^{3} + 8 x^{2} + 6 x - 2} = \lim_{x \to \infty}\frac{6 x^{2} + 4 x - 7}{6 x^{2} + 16 x + 6} = \lim_{x \to \infty}\frac{4 \left(3 x + 1\right)}{4 \left(3 x + 4\right)} = \lim_{x \to \infty}\frac{12}{12} = 1