One does not need long division here. In any case, I have not seen long division applied to non-polynomials, though there may be a way to extend it.

In rate of growth problems, log(x) is a little brother of x. Also, the rate of a polynomial is determined by its degree. Informally, the nominator here is dominated by , which is approximately . The denominator is also approximately , so the whole function is O(1).

More formally, you can write a series of upper bounds, e.g.: . Some of those inequalities may not hold for all x, but only for x > k for some k.