Evaluate the limit LaTeX: \displaystyle \lim_{x \to \infty}\frac{7 x^{3} + 9 x^{2} + 5 x + 6}{9 x^{3} - 6 x^{2} + 9 x + 3}

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} + 9 x^{2} + 5 x + 6}{9 x^{3} - 6 x^{2} + 9 x + 3} = \lim_{x \to \infty}\frac{21 x^{2} + 18 x + 5}{27 x^{2} - 12 x + 9} = \lim_{x \to \infty}\frac{6 \left(7 x + 3\right)}{6 \left(9 x - 2\right)} = \lim_{x \to \infty}\frac{42}{54} = \frac{7}{9}