
Limit to infinity
$\displaystyle \lim_{x \to \infty} \frac{\sqrt{9x^6  x}}{x^3 + 1}\\
\frac{\sqrt{9x^6  x}}{x^3 + 1} \cdot \frac{\frac{1}{x^3}}{\frac{1}{x^3}} = \\
\frac{\frac{\sqrt{9x^6  x}}{x^3}}{1 + \frac{1}{x^3}} = \\
\frac{(1 + \frac{1}{x^3})(\sqrt{9x^6  x})}{x^3} = \\
\frac{(1 + \frac{1}{x^3})(\sqrt{9x^6  x})}{x^3} \cdot \frac{\frac{1}{x^3}}{\frac{1}{x^3}} = \\
\frac{(1 + \frac{1}{x^3})(\sqrt{9x^6  x})}{x^3}$
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?

Re: Limit to infinity
Take your denominator into the square root $\displaystyle \sqrt{\frac{9x^6x}{(x^3+1)^2}}$.
You may see that the numerator and denominator both have 6 as their highest power of x.
Divide top and bottom by x^6.