On the theorem of infinitesimal and infinitesimal For example, theorem: the sum of finite infinitesimals is also infinitesimal Suppose that when x tends to x0, it is proved in the book that the sum of two infinitesimals when x tends to x0 satisfies the condition of infinitesimal But I think, why do two infinitesimals just tend to x0 when proving? The theorem says that two infinitesimals do not necessarily have the same x0? For example, the x power of (0.5) and the x power of 2 are infinitesimal. Although these two functions are not infinitesimal when they tend to finite values, their sum will not be infinitesimal Why?

On the theorem of infinitesimal and infinitesimal For example, theorem: the sum of finite infinitesimals is also infinitesimal Suppose that when x tends to x0, it is proved in the book that the sum of two infinitesimals when x tends to x0 satisfies the condition of infinitesimal But I think, why do two infinitesimals just tend to x0 when proving? The theorem says that two infinitesimals do not necessarily have the same x0? For example, the x power of (0.5) and the x power of 2 are infinitesimal. Although these two functions are not infinitesimal when they tend to finite values, their sum will not be infinitesimal Why?

It is a misunderstanding that two infinitesimals of the theorem do not necessarily have the same x0
Infinitesimal is limit in essence, and limit has limit process. If two limits can be operated, their limit process must be the same. Otherwise, the theorem that "the sum of finite infinitesimals is also infinitesimal" does not hold
As you mentioned, although the x power of (0.5) and the x power of 2 can be regarded as infinitesimals, they are infinitesimals of different processes. Therefore, if the operation is carried out, it will inevitably lead to the same process, that is, two can not be infinitesimals at the same time. Of course, the addition is not infinitesimal