The limit of a variable

The limit of a variable

We have called a variable an infinitesimal, if its corresponding point K in the axis X has on displacement the following property: on successive variation the length of the interval OK becomes, and on further variation remains, less than any given positive number . We now suppose that this property is fulfilled, not by the interval OK, but by AK, where A is a definite point on the axis X with abscissa a (Fig. 42).

In this case, the interval S'S of length 2 will have its center at the point A, abscissa x = a, instead of at the origin, and the point K must come within this interval on successive displacement, then remain there on further displacement. We say in this case that the constant a is the limit of variable x, or that x tends to a.

Noting that the length of AK is | a-x |, we can formulate the following definition:

DEFINITION. The constant a is called the limit of the variable x when the difference a-x (or x-a) is an infinitesimal, Having regard to the definition of an infinitesimal, a limit can be thus defined:

DEFINITION. The constant a is called the limit of the variable x, when we have the following property : for any given positive  there exists a value of x such that, for all subsequent values, |a-x| < .

We note some immediately obvious consequences of this definition, without dwelling on their detailed proof.

No variable can tend to two different limits, and not every variable has a limit. for example, the variable sin(a) oscillates between -1 and 1 on successive increase of the angle a, and has no limit. The limit of an infinitesimal is zero.

if x and y vary simultaneously, and each tends to a limit in the course of successive variation, whilst both always satisfy x <= y, their limits a and b satisfy the condition a <= b. We note here, that if the variables satisfy x < y, the sign of equality can be obtained for their limits, i.e. we have a <= b. If x, y, z vary simultaneously and always satisfy the condition x <= y <= z on successive variation, and if x and z tend to the same limit a, y also tends to the limit a.

If a is the limit of x (or x tends to a), we write:

If x tends to a, the difference x-a =  is an infinitesimal, and we can write:

i.e. every variable tending to a limit can be expressed as the sum of two terms: a constant term, equal to the limit, and an infinitesimal. Conversely, if a variable x can be expressed in the form (2), where a is a constant, and  is an infinitesimal, the difference x-a will be an infinitesimal, and hence, a is the limit of x.

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek