Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{7 x^{3} - 5 x^{2} + 5 x + 8}{9 x^{3} - 2 x^{2} - 8 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} - 5 x^{2} + 5 x + 8}{9 x^{3} - 2 x^{2} - 8 x + 1} = \lim_{x \to \infty}\frac{21 x^{2} - 10 x + 5}{27 x^{2} - 4 x - 8} = \lim_{x \to \infty}\frac{2 \left(21 x - 5\right)}{2 \left(27 x - 2\right)} = \lim_{x \to \infty}\frac{42}{54} = \frac{7}{9}