Basic theorems of limits

Basic theorems of limits

1-The limit of the algebraic sum of a finite number of variables is equal to the sum of their limits.

For the sake of exactness let us take the algebraic sum x-y+z of three simultaneously varying magnitudes. We suppose that x, y and z tend respectively to limits a, b and c. We show that the sum tends to the limit a - b + c.

We have:

where  are infinitesimals. We can write for the sum:

The first bracket on the right-hand side of this equation is a constant, and the second is an infinitesimal. Hence:

2-The limit of the product of a finite number of variables is equal to the product of their limits.

We confine ourselves to the case of the product x*y of two variables. We suppose that x and y vary simultaneously, tending respectively to limits a and b, and we show that x*y tends to the limit a*b.

We have by hypothesis:

where  are infinitesimals; hence:

Using both of the properties of infinitesimals, we see that the sum in the bracket on the right of this equation is an infinitesimal, and hence we have:

3-The limit of a quotient is equal to the quotient of the limits, provided the limit of the denominator is not zero.

We take the quotient x/y, and suppose that x and y tend simultaneously to their respective limits a and b, where  We show that x/y tends to a/b.

To prove the theorem, it is sufficient to show that the difference a/b - x/y is an infinitesimal. By hypothesis:

where  are infinitesimals. Hence:

The denominator of the fraction on the right of this equation is the product of two factors, and tends to b^2. Thus, from some initial moment of its variation, it is greater than b^2/2, the fraction as a whole being included between zero and 2/b^2, i.e. the fraction is bounded. The term  is an infinitesimal. Hence, the difference a/b-x/y is an infinitesimal, and

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A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek

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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

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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

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Ordered variables

Ordered variables

When referring to the independent variable x, we have only been concerned with the set of the values that x can assume. For example, this can be the set of values satisfying 0 < x < 1. We shall now consider the variable x taking an infinity of values in sequence, i.e. we are now interested, not only in the set of values x, but also in the order in which it takes these values. More precisely, we assume the possibility of distinguishing, for every value of x, a value that precedes it and a value that follows it, it being also assumed that no value of the variable is the last, i.e. whatever value we take, there exists an infinity of successive values.

A variable of this type is sometimes called ordered. If x', x" are two values of the ordered variable x, a preceding and a succeeding value can be distinguished, whilst if x' precedes x", and x" precedes x'", then x' precedes x"' . We shall assume, for example, that the set of values of x is defined by 0 < x < 1, and that of two distinct values x' and x" the succeeding value is the greater. We thus obtain an ordered variable, continuously increasing through all real values from zero to unity, without reaching unity. The sequence of values of the variable, for phenomena occurring in time, is established by the temporal sequence, and we shall sometimes make use of this time-scheme below, using terms such as "previous" and "later" in place of "preceding" and "succeeding" values.

An important particular case of an ordered variable is that when the sequence of values of the variable can be enumerated, by arranging them in a series of the form:

so that, given two values the value succeeds that has the greater subscript. In the case mentioned above, when the variable increases from zero to unity, we can clearly not enumerate its successive values. It may also be noted that it is possible to encounter identical values amongst those of an ordered variable. For example, we might have  in the enumerated variable. Abstracting, as we always do, from the concrete nature of the magnitude (length, weight etc.), we must understand by the term "ordered variable", or as we shall say for brevity, "variable", simply the total sequence of its numerical values. We normally introduce one letter, say x, and suppose that it assumes successively the above-mentioned numerical values.

For every value of the variable x, a corresponding point K is defined on the axis X. Thus, as x varies in sequence, the point K moves along X. The present paragraph is devoted to the basic theory of limits, which is fundamental to all modern mathematical analysis. This theory considers some extremely simple, and at the same time, extremely important, cases of variation of magnitudes.

 the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek

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Inverse trigonometric, or circular, functions

Inverse trigonometric, or circular, functions

These functions are obtained by inversion of thetrigonometric functions:

their symbols being respectively:

these symbols are simply abbreviated forms of description for the angle (or arc), of which the sine, cosine, tangent or cotangent is respectively equal to x.

We shall consider the function:

The graph of this function (Fig. 39) is obtained from the graph of y = sin x by the rule given in Inverse functions. This graph is wholly located in the vertical strip of width two, based on the interval - 1 < x < + 1 of the axis X, i.e. the function (22) is only defined in the interval -1 < x <+1. Furthermore, equation (22) is equivalent to the equation sin y = x; and, as is known from trigonometry, for a given x we obtain an infinite number of values of y. We see from the graph, in fact, that perpendiculars to the axis X from points in the interval -1 < x < +1 have an infinite number of points in common with the graph, i.e. function (22) is many-valued.

We see directly from Fig. 39 that function (22) becomes single valued if, instead of taking all the graph, we limit ourselves to the part shown in heavier type, which corresponds to stipulating that we shall consider only those values of the angle y, having a given sin y = x, which lie in the interval

Figures 40 and 41 illustrate the graphs of y= arc cos x and y= arctan x, the parts of the graphs in heavier type being those which must be kept in order to make the functions single-valued (we leave it to the reader to draw the figure for arc cot x). It may be noted here that the functions y = arc tan x and y = arc cot x are defined for all real values of x.

By noting from the figure, the interval of variation of y over the heavier part of the curve, we obtain a table of bounds, within which the function remains single-valued:

It can easily be shown that the functions thus defined, called the principal values of the inverse trigonometric functions, satisfy the relationships:

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek

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