Infinitesimals (to understanding limit theory)

Infinitesimals (to understanding limit theory)

We assume that the point K constantly remains inside a certain interval of the axis X. This is equivalent to the condition that the length of the interval 0X, where 0 is the origin, remains less than a definite positive number M. The magnitude x is said to be bounded in this case. Noting that the length OK is |x|, we can give the following definition:

A variable x is said to be bounded, if there exists a positive number M, such that |x|< M for all values of x. We can take x = sin(a) as an example of a bounded magnitude, where the angle a varies in any manner. Here, M can be taken as any number greater than unity.

We now consider the case when the point K is displaced successively, and indefinitely approaches the origin. More precisely, we suppose that successive displacements of point K bring it inside any previously assigned small section S'S of the axis X with center 0, and that it remains inside this section on further displacement. In this case, we say that the magnitude x tends to zero or is an infinitesimal.

We denote the length of the interval S'S by 2, where  signifies any given positive number. If the point K is inside S'S, then OK <  and conversely, if OK <, K is inside S'S. We can thus give the following definition: The variable x tends to zero or is an infinitesimally if for any given positive there exists a value of x, such that for all subsequent values of x, |x| < .

In view of the importance of the concept of infinitesimal, we give another formulation of the same definition. 

DEFINITION: A magnitude x is said to tend to zero or to be an infinitesimal, if on successive variation |x| becomes, and on further variation remains, less than any previously assigned small positive number .

The term "infinitesimal" denotes the character of the variation of the variable described above, and the underlying concept is not to be confused with that of a very small magnitude, which is often

employed in practice.

Suppose that, in measuring a certain tract of land, we obtained 1000 m, with some remainder that we considered very small in comparison with the total length, so that we neglected it. The length of this remainder is expressed by a definite positive number, and the term "infinitesimal" is evidently not applicable here. If we were to meet with the same remainder in a second, more accurate measurement, we should cease to consider it as very small, and we should take it into account. It is thus clear that the concept of a small magnitude is a relative concept, bound up with the practical nature of the measurement.

Suppose that the successive values of the variable x are

and let   be any given positive number. To prove that x is an infinitesimal, we must show that, starting with a certain value of n, |xn| will be less than , i.e. we must be able to find a certain integer N such that

This N depends on .

As an example of an infinitesimal, we take the magnitude assuming successively the values:

.

We have to satisfy the inequality:

Remembering that  is negative, we can rewrite the above inequality as:

since division by a negative number changes the sense of the inequality; thus we can now take N as the largest integer in the quotient  ? Thus the magnitude in question, or as we usually say, the sequence (1) tends to zero.

If we replace q by (-q) in the sequence (1), the only difference is the appearance of the minus sign with odd powers; the absolute magnitude of the members of the sequence is as before, and hence we also have an infinitesimal in this case. 

The fact that x is infinitesimal is usually denoted by:

Here, lim is an abbreviation of ''limit''.

 the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek