Specifically, if a change trend x->; One,
1. If lim f(x)/g(x)=0, then f (x) = o (g (x));
2. If m > exists; 0 makes | f (x)/g (x) |
There are some similar marks, such as
3.if | f(x)/g(x)| & gt； = M>0, then f (x) = ω (g (x))
4. If it is 0
5. If lim f(x)/g(x)= 1, then f (x) ~ g (x) are equivalent quantities.
Generally speaking, the small O symbol is only used for infinitesimal, and the large O symbol is used for the comparison between infinitesimal and infinitesimal. In addition, pay attention to the changing trend (such as X->; A) It can only be omitted if it does not cause misunderstanding.
As for the operation rules, there is no need to summarize and analyze the specific situation. If you encounter specific problems that cannot be solved, it means that you are not learning well, so even if you recite some rules, it is useless.
For example, o(u)+o(v)=o(|u|+|v|), o(u)o(v)=o(uv), which are just a layer of encapsulation defined, are basically of little value. If x->; O(x)+O (x 2) is 0, and you should know that the result is O (x), which is not very dogmatically written as O (| x |+| x | 2).
As for the transformation between small O and large O, we can get o(u)=O(u) directly from the definition, but on the other hand, there is no universal conclusion. Under certain conditions, x->; When 0, O (x k) can be changed into O (x {k+ 1}), for example, the remainder of the n-order Maclaurin expansion of k+ 1 differentiable function has these two forms. However, it is generally not true, such as x->; 0 x/lnx=o(x), but it cannot be changed to o (x 2).