Yup, doing a big of handwavy math also demonstrates this: You are comparing something like x^N - x^M with 2^x - x^M. The second will always grow big rapidly, wh

Yup, doing a big of handwavy math also demonstrates this: You are comparing something like x^N - x^M with 2^x - x^M. The second will always grow big rapidly, while the first could stay small, but it can also grow arbitrarily large. In fact, it is bound to do so for N > M and it will do so quite rapidly if N >> M.