Infinitesimo. An infinitesimal, infinitely small quantity, used in infinitesimal calculus , are defined as limits and considered as numbers in practice.
Summary
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- 1 Definition
- 1 Examples
- 2 Properties
- 3 infinitesimal comparables
- 4 infinitely many equivalents
- 5 Definition of order of an infinitesimal
- 6 Sources
Definition
A function y = f (x) is infinitely small, infinitesimal or infinitesimal when x → a (or when x → ∞) if and only if
From the definition it is deduced whether
Then for any number ε, however small, there exists an environment of radius δ ( a – δ, a + δ) such that for each x that belongs to ( a – δ, a + δ) it is verified that ǀf (x) ǀ <ε.
Yes
So for any number ε, however small it is, there exists a number x 0 that belongs to the set of real numbers such that for each x> x 0 it is verified that ǀf (x) ǀ <ε.
That is to say, an infinitesimal is a function whose limit is zero when the independent variable x approaches the value x = a , or in other words, a function whose values get closer to zero the closer x gets to the value a . Therefore, in the concept of infinitesimal one must keep in mind not only the function f, but also the point a . The function f is infinitesimal, in the vicinity of point a . It is often said to be infinitesimal at x = a .
Examples
The following functions are infinitesimal at x = 0,
- f (x) = x
- g (x) = 1-cos (x)
The following functions are infinitesimal,
- h (x) = 1 / x when x → ∞
- k (x) = sin x when x → 0
