3.2.3 Asymptotic Properties Video
PROFESSOR: An advantage of expressing the asymptotic notations, in terms of limits, is that a bunch of their properties then become immediately obvious. Here's one. If f is little o of g, or f is asymptotically equal to g, then, in fact, f is big o of g. Or we can reason about this informally by saying that the first one means that f is much less than g, and the second one means that f is about the same as g, and the final one means that f is roughly less. So being about the same and definitely less, and certainly this implies roughly less. But we can in fact be entirely precise just using the definitions, because f equals o of g means the limit of f over g is 0. And f is asymptotically equal to g means that the limit of f over g is 1. And the definition of f equals big o of g is that the limit is finite. And clearly, if it's 0 or 1, then it's finite. Another such property is that if f is much less than g, then g is not roughly less than f. More precisely, if ...