$\displaystyle \lim_{x \to \infty} \frac{\sqrt{9x^6 - x}}{x^3 + 1}\\

\frac{\sqrt{9x^6 - x}}{x^3 + 1} \cdot \frac{\frac{1}{x^3}}{\frac{1}{x^3}} = \\

\frac{\frac{\sqrt{9x^6 - x}}{x^3}}{1 + \frac{1}{x^3}} = \\

\frac{(1 + \frac{1}{x^3})(\sqrt{9x^6 - x})}{x^3} = \\

\frac{(1 + \frac{1}{x^3})(\sqrt{9x^6 - x})}{x^3} \cdot \frac{\frac{1}{x^3}}{\frac{1}{x^3}} = \\

\frac{(1 + \frac{1}{x^3})(\sqrt{9x^6 - x})}{x^3}$

And it just repeats over and over again and I can't find anything to divide by without destroying the work I've already done. What am I supposed to do in a loop and there's nothing to divide by?