[devnulloverflow]

> The author instead seems to think they are approximately the same because the lower order terms are small compared to the highest one.

In fairness, this is actually why we care in practice about asymptotic issues. The fact that there is a rigorous mathematical viewpoint that makes them strictly comparable is helpful for being confident about the non-trivial results -- but the reason we care about it is because we want to know roughly how expensive our code is (or will be).