Infinite functions add up, which is naturally indefinite.
Of course, this function can be a constant function-the only constant infinitesimal is f(x)≡0. But on the whole, we are considering the infinitesimal value.
In fact, the multiplication of an infinite number of functions with a limit of 0 is also undetermined. Here is an example, the product of infinite infinitesimals is infinite:
For any positive integer n, consider the function fn(x) whose domain is [1, +∞).
Obviously, for any positive integer n, when x→+∞, the limit of fn(x) is 0, which is the so-called "infinitesimal". But consider the product of all fn(x)
Note that for any x∈[ 1, +∞), let x∈[k, k+ 1], where k is a positive integer, then
So it is easy to know that F(x)=x is infinite when x→+∞.
Author: Shen Lili
Link:/question/24953088/answer/29787272
Source: Zhihu.