Here we expressly remark that this includes the trivial
case in which all the numbers an are
equal to one another, and hence also coincide with the limit. 1.6 Further Discussion of the Concept of Limit
1.6.1 First Definition of Convergence: The examples discussed in the last section guide us to the general concept of limit:
If an infinite sequence of
numbers a1, a2, a3,
··· , an, ··· is given and if
there is a number l such that every
interval, however small, marked off about the point l,
contains all the points an, except
for at most a finite number, we say that the number l
is the limit of the sequence a1, a2,
··· or that the sequence a1, a2,
··· converges to l; in symbols, Here we expressly remark that this includes the trivial
case in which all the numbers an are
equal to one another, and hence also coincide with the limit.
Instead of the above, we may use the following equivalent statement: If any positive number e be assigned - however small - one can find a whole number N=N(e) such that from N onward (i.e., for n > N(e)), it is always true that |an-l| < e. Of course, it is as a rule true that the bound N(e) will have to be chosen larger and larger with smaller and smaller values of e ; in other words, N(e) will increase beyond all bounds as e tends to 0.
It is important to remember that every convergent sequence is bounded, that is, there corresponds to every sequence a1, a2, a3, ···, for which a limit l exists, a positive number M, independent of n, such that for all the terms an of the sequence the inequality |an| < M is valid.
This theorem readily follows from our definition. We choose e equal to 1; then there is an index N such that |an - l| < 1 for n > N. Let A be the largest among the numbers
We can then put M = [l]+ A + l. Since, by the definition of A, the inequality |an-l|< A + 1 certainly holds for n= 1, 2, ··· , N, while for n > N
A sequence which does not converge is said to
be divergent. If, as n increases, the numbers an increase beyond all bounds, we say that the sequence
diverges to +¥ and, as we have already done occasionally, we write 
Similarly, we write 
, if, as n increases, the numbers -
an increase beyond all bounds in the positive direction.
But divergence may manifest itself in other ways, as, for
example, in the case of the sequence 
the terms of which
oscillate between two different values.*
*Another useful remark: As regards convergence, the behaviour of a sequence is unaltered, if we omit a. finite number of the terms an. In what follows, we shall frequently use this, speaking of the convergence or divergence of series in which the term an is undefined for a finite number of values of n.
In all the examples given above, the limit of the sequence considered is a known number. If the concept of limit were to yield nothing more than the recognition that certain known numbers can be approximated as closely as we like by certain sequences of other known numbers, we should have gained very little. The fruitfulness of the concept of limit in analysis rests essentially on the fact that limits of sequences of known numbers provide a means of dealing with other numbers, which are not directly known or expressible. The whole of higher analysis consists of a succession of examples of this fact, which will become steadily clearer to us in the following chapters. The representation of the irrational numbers as limits of rational numbers may be regarded as a first example. In this section, we shall become acquainted with further examples. However, before we take up this subject, we shall make a few general, preliminary remarks.
1.6.2 Second (Intrinsic) Definition of Convergence: How can we tell that a given sequence of numbers a1, a2, a3, ···, an, ··· converges to a limit, even when we do not know beforehand what that limit is? This important question is answered by Cauchy's convergence test.*
It is sometimes referred to as the general principle of convergence.
We say that a sequence of numbers a1, a2, a3, ···, an, ··· is convergent if there corresponds to every arbitrarily selected small positive number e a number N=N(e), which usually depends on e, such that |an - am| < e, provided that n and m are both at least equal to N(e ). Cauchy's convergence test can then be expressed as follows:
Every intrinsically convergent sequence of numbers possesses a limit.
The importance of Cauchy's test lies in the fact that it allows to speak of the limit of a sequence after considering the sequence itself without any further information about the limit itself. The converse of Cauchy's test is very easy to prove. For, if the sequence a1, a2, ··· tends to the limit l, then, by the definition of convergence, we have
where e is a positive quantity as small as we please, provided only that both m and n are large enough, whence
Since e can be chosen as small as we please, this inequality expresses our statement.
Cauchy's test itself becomes intuitively obvious, if we think of the numbers as they are represented on the number axis. It then states that a sequence certainly has a limit, if after a certain point N all the terms of the sequence are restricted to an interval which can be made arbitrarily small by choosing N large enough.
In Appendix 1, we shall show how Cauchy's test can be proved by purely analytical methods. For the time being, we will accept it as a postulate.
1.6.3 Monotonic Sequence: The question whether a given sequence converges to a limit is particularly easy to answer when the sequence is a so-called monotonic sequence, that is, if either every number of the sequence is larger than the preceding number (monotonic increasing) or else every number is smaller than the preceding number (monotonic decreasing). We have the theorem:
Every monotonic increasing sequence the terms of which are bounded above (that is, lie below a fixed number) possesses a limit; similarly, every monotonic decreasing sequence the terms of which never drop below a certain fixed bound possesses a limit.
For the present, we shall regard these results as obvious, merely referring the student to the rigorous proof in Appendix 1. Naturally, a convergent, monotonic, increasing sequence must tend to a limit which is greater than any term of the sequence, while for a convergent monotonic decreasing sequence the numbers tend to a limit which is smaller than any number of the sequence. Thus, for example, the numbers l/n form a monotonic decreasing sequence with the limit 0, while the numbers 1 - l/n form a monotonic increasing sequence with the limit 1.
In many cases, it is convenient to replace the condition that a sequence shall increase monotonically by the weaker condition that its terms shall never decrease, in other words, to allow its successive terms to be equal to one another. We then speak of a monotonic non-decreasing sequence, or of a monotonic increasing sequence in the wider sense. Our theorem on limits remains true for such sequences as well as for sequences which are monotonic non-increasing or monotonic decreasing in the wider sense.
1.6.4 Operations with Limits: We conclude with a remark concerning calculations with limits. It follows almost at once from the definition of limit that we can perform the elementary operations of addition, multiplication, subtraction, and division according to the rules:
If a1, a2, ··· is a sequence with the limit a and b1, b2, ··· a sequence with the limit b, then the sequence of numbers cn = an + bn also has a limit and
The sequence of numbers cn = an·bn likewise converges, and
Similarly, the sequence cn = an - bn converges, and
Provided the limit b differs from 0, the numbers cn = an/bn also converge, and have the limit
In words: we can interchange the rational operations of calculation with the process of forming a limit, that is, we obtain the same result whether we first perform a passage to the limit and then a rational operation or vice versa.
For the proof of these simple rules, it is sufficient to give one example; using this as a model, the reader can establish the other statements by himself. As this example, we consider multiplication of limits. The relations an ® a and bn ® b amount to the following: If we choose any positive number e, we need only take n greater than N, where N = N(e) is a sufficiently large number depending on e, in order to have both
If we write ab - anbn = b(a - an)+ an(b - bn) and recall that there is a positive bound M, independent of n, such that |an| < M, we obtain
Since the quantity ( |b| + M)e can be made arbitrarily small by choosing e small enough, we see that the difference between ab and anbn actually becomes as small as we please for all sufficiently large values of n, which is precisely the statement made in the equation
By means of these rules, many limits can be evaluated very easily; for example, we have
since in the second expression the passage to the limit in the numerator and denominator can be made directly.
Another simple and obvious rule is worth stating. If lim an = a and lim bn = b, and if, in addition, an > bn for every n, then a ³ b. However, we are by no means entitled to expect that, in general, a will be greater than b, as is shown by the case of the sequences an = l/n, bn = l/2n, for which a = 0 = b.
1.6.5 The Number e: As a first example of the generation of a number, which cannot be stated in advance as the limit of a sequence of known numbers, we consider the sums
We assert that, as n increases, Sn tends to a definite limit.
In order to prove the existence of the limit, we observe that, as n increases, the numbers Sn increase monotonically. For all values of n, we also have
Hence the numbers Sn have the upper bound 3 and, being a monotonic increasing sequence, they possess a limit, which we denote by e:
Moreover, we assert that the number e, defined as the above limit, is also the limit of the sequence
The proof is simple and at the same time an instructive example of operations with limits. According to the binomial theorem, which we shall here assume,
Hence we see at once: (1) that Tn
£ Sn,
and (2) that the Tn also form a
monotonic increasing sequence*, whence there follows the
existence of the limit 
. In order to prove that T = e,
we observe that
provided that m > n. If we
now keep n fixed and let m increase beyond all
bounds, we obtain on the left hand side the number T and
on the right hand side the expression Sn,
so that T ³ Sn. We have thus
established the relationship T ³ Sn
³ Tn
for every value of n. We can now let n
increase, so that Tn tends to T;
it follows from the double inequality that 
as was to be proved.
* We obtain Tn+1 from Tn by replacing the factors 1-1/n, 1-2/n,··· by the larger factors 1-1/(n+1), 1-1/(n+1), ··· and finally adding a positive term.
We shall later meet this number e again from still another point of view.
1.6.6 The number p as a limit: A limiting process, which in essence goes back to classical antiquity (Archimedes), is that by which the number p is defined. Geometrically speaking, p means the area of the circle of radius 1. We therefore accept the existence of this number as intuitive, regarding it as obvious that this area can be expressed by a (rational or irrational) number, which we then simply denote by p. However, this definition is not of much help to us, if we wish to calculate the number with any accuracy. We have then no choice but to represent it by means of a limiting process, namely, as the limit of a sequence of known and easily calculated numbers. Archimedes himself used this process in his method of exhaustions, where he steadily approximated the circle by means of regular polygons with an increasing number of sides fitting it more and more closely. If we denote by f the area of the regular polygon with m sides, inscribed in the circle, then the area of the inscribed 2m polygon is given by the formula (proved by elementary geometry)
We now let m run, not through the sequence of all positive integers, but through the sequence of powers of 2, that is, m = 2n; in other words, we form those regular polygons the vertices of which are obtained by repeated bisection of the circumference. The area of the circle is then given by the limit
In fact, this representation of p as a
limit serves as a base for numerical computations; starting with
the value f4 = 2, we can calculate in order
the terms of our sequence tending to p. An estimate of
the accuracy with which any term 
represents p can be
obtained by constructing the lines touching the circle and
parallel to the sides of the inscribed 2n
polygon. These lines form a circumscribed polygon, similar to the
inscribed 2n polygon, and have greater
dimensions in the ratio 1: cosp/2n-1, whence the area 
of the
circumscribed polygon is given by
Since the area of the circumscribed polygon is evidently larger than that of the circle, we have
These are matters with which the reader will be more or less familiar. What we wish to point out here is that the calculation of areas by means of exhaustion by rectilinear figures, the areas of which are readily calculated, forms the basis for the concept of integral to be introduced in the next chapter.
1.* (a)
Replace the statement the sequence an
is not absolutely bounded by an equivalent
statement not involving any form of the words bounded or unbounded.
(b) Replace the statement the sequence an is
divergent by an equivalent statement not
involving any form of the words convergent or divergent.
2.* Let a1 and b1 be two positive numbers and a1 < b1. Let a2 and b2 be defined by the equations
Similarly, let
and, in general,
Prove (a) that the sequence an a1, a2, ··· converges, (b) that the sequence b1, b2, ··· converges, (c) that the two sequences have the same limit. (This limit is called the arithmetic-geometric mean of a1 and b1.)
3.* Prove that if 
then 
, where sn
is the arithmetic mean 
4. If 
, shows that the arithmetic means of the
arithmetic means sn tend towards x
5. Find the error involved in using 
as an
approximation to e. Calculate e accurately to 6
decimal places.
1.7 The Concept of Limit where the Variable is Continuous
Hitherto, we have considered limits of sequences, that is, of functions of an integral variable n. However, the notion of limit frequently occurs in connection with the concepts of a continuous variable x and a function.
We say that the value of the function f(x) tends to a limit l as x tends to x, or in symbols
if all the values of the function f(x), for which x lies near enough to x, differ arbitrarily little from l. Expressed more precisely, the condition is:
If an arbitrarily small positive quantity e is assigned, we can mark off about x an interval |x - x| < d so small that there applies for every point x in this interval different from x itself the inequality |f(x) - l| < e .
We expressly exclude here the equality of x and x. This is done purely for reasons of expediency, so as to have the definition in a form more convenient for applications, e.g., in the case where the function f(x) is not defined at the point x, although it is defined for all other points in a neighbourhood of x.
If our function is defined or considered in a given interval only, for example,
we shall restrict the values of x to this interval. Thus, if x denotes an end-point of an interval, x is made to approach x by values on one side of x only (limit from the interior of the interval or one-sided limit).
As an immediate consequence of this definition, we have the fact:
If 
and x1,
x2, ··· , xn,
··· is a sequence of numbers all different from x , but approaching x as a limit, then 
.
In fact, let e be any positive number; we wish to show that for all values of n greater than a certain n0 there applies the inequality
By definition, there exists a d > 0 such that, whenever |x - x| < d , one has the inequality
Since xn ® x, the relation |x - x| < d is satisfied for all sufficiently large values of n, and it follows for such values that |f(xn) - l| < e, as was to be proved.
We shall now attempt to clarify this abstract definition by means of simple examples. Consider first the function
defined for x ¹ 0. We state that
We cannot prove this statement simply by carrying out the passage to the limit in the numerator and denominator separately, because the numerator and denominator vanish when x = 0, and the symbol 0/0 has no meaning. We arrive at the proof as follows:
From Fig. 18, we find by comparing
the areas of the triangles OAB and OAC and the
sector OAB that, if 0<x<p/2,
From this follows that, if 0<|x|<p/2,
Hence the quotient sin x/x lies between the numbers 1 and cos x . We know that cos x ® 1 as x® 0, whence the quotient sin x/x can only differ arbitrarily little from 1, provided that x is near enough to 0. This is exactly what is meant by the equation which was to be proved.
By the result just proved
and also
This last result follows from the formula, valid for 0 < |x| < p/2,
As x ® 0, the first factor on the right hand side tends to 1, the second one to 1/2 and the third one to 0, whence the product tends to 0, as has been stated.
The same formula, on division by x, yields
whence
Finally, consider the function 
, defined
for all values of x. This function is never negative; it
is equal to x for x ³ 0 and to - x
for x < 0. In other words, 
Hence the function 
defined
for all non-zero values of x, has the value +1 when x
> 0 and —1 when x < 0, whence the limit
cannot exist, since arbitrarily near to 0 we can find values of x
for which the quotient is +1 and other values for which it is -1.
In concluding this discussion on limits in connection with a continuous variable, we remark that it is, of course, possible to consider limiting processes in which the continuous variable x increases beyond all bounds. We state the example
without further discussion. It signifies that the function on the left hand side differs arbitrarily little from 1, provided only that x is sufficiently large.
In these examples, we have proceeded as if operations with limits obeyed the same laws in the case of continuous variables as in the case of sequences. That this is actually true, the reader himself can verify; the proofs are essentially the same as for limits of sequences.
Exercises 1.6: 1. Find the following limits, giving at each step the theorem on limits which justifies it:
2. Prove that
3. Determine whether or not the following limits exist and, if they do exist, find their values:
1.8. The Concept of Continuity
1.8.1 Definitions: We have already illustrated in 1.2.3 by means of examples the notion of continuity. Now, using the idea of limit, we can make the concept of continuity precise.
We thought of the
graph of a function, which is continuous in an interval, as a
curve consisting of one unbroken piece; we also stated that the
change in the function y must remain arbitrarily small,
provided only that the change of the independent variable x
is restricted to a sufficiently small interval. This state of
affairs is usually formulated as follows, with a greater range
but increased precision. A function f(x) is
said to be continuous at the point x, if it
possesses the property: At
the point x,
the value of the
function f(x)
is approximated to within an arbitrary pre-assigned degree of
accuracy e by all functional values f(x)
for which x is near enough to x. In other words, f(x)
is continuous at x,
if for every positive number e, no matter how small, there can be determined another
positive number d = d(e)
such that |f(x) - f(x)| < e (Fig. 19) for all points x for which |x -
x| < d. Or
again: The condition of
continuity requires that for the point x
The value of the function at the point x is the same as the limit of the functional values f(xn) for any arbitrary sequence xn of numbers converging to x.
It is important to observe that our condition
involves two different matters: (1) the existence of 
and (2) the coincidence of this limit with f(x), the value of the
function at the point x.
Having now defined continuity of a function f(x) at a point x, we proceed to state what we mean by the continuity of a function f(x) in an interval. This may be defined simply as follows: The function f(x) is continuous in an interval, if it is continuous at each point of that interval. Stated fully, this requires that, if a positive number e be assigned, then there exists for each point x of the interval a number d, depending as a rule on e and on x, such that
and 
lies in the interval a
£ £ b.
Closely related to this is another concept, namely that of uniform continuity. A function f(x) is uniformly continuous in the interval a £ x £ b, if there exists for every positive number e a corresponding positive number d such that, for every pair of points x1, x2 in the interval, the distance |x1 - x2| a part of which is less than d, one has the inequality |f(x1) - f(x2)| < e. This differs from the definition stated above in that d in the definition of uniform continuity does not depend on x, but is equally effective for all values of x, whence follows the term uniform continuity.
It is quite obvious that a uniformly continuous function is necessarily continuous. Conversely, it can be shown that every function f(x), which is continuous in a closed interval a £ x £ b, is also uniformly continuous. The proof of this is given in Appendix I. Even though the reader may not desire to read the proof at present, he will find it helpful to study the examples at the beginning of A1.2.2. But until the student has worked through this proof, he may assume, whenever a function is said to be continuous in a closed interval, uniform continuity is implied.
1.8.2 Concept of Discontinuity: We can understand the concept of continuity better if we study its opposite, the concept of discontinuity. The simplest discontinuity occurs at those points at which a function has a jump, that is, at which the function has a definite limit as x tends to the point from the right and a definite limit as x tends to the point from the left, while these two limits are different. It does not matter whether or how the function is defined at the point of discontinuity itself.
For example, the function f(x) defined by the equations
has discontinuities at the points x = 1 and x = -l. The limits on approaching these points from the right and from the left hand side differ by 1, and the values of the function at these points agree with neither limit, but are equal to the arithmetic mean of the two limits.
It may be noted in passing that our function can be represented, using the idea of a limit, by the expression
For, if x² < 1, that is, if x lies in the interval -1 < » < 1, the numbers x2n will have the limit 0, and the function will have the value 1. However, if x² > 1, as n increases, x2n will increase beyond all bounds; our function will then have the value 0. Finally, obviously, for x2 = 1, that is, for x = +1 and x = -1, the value of the function is ½ (Fig. 20).
;
Other curves with jumps are shown in Figs. 21a and 21b; they represent functions with obvious discontinuities.
In the case of discontinuities of this kind, both the limits from the right and from the left exist. We now proceed to the consideration of discontinuities in which this is not the case. These most important of such discontinuities are the infinite discontinuities or infinities. These are discontinuities which are exhibited by the functions 1/x or l/x² at the point x = 0; as x ® x, the absolute value | f(x)| of the function increases beyond all bounds. In the case of 1/x, the function increases numerically beyond all bounds through positive and through negative values, respectively, as x approaches the origin from the right and from the left. On the other hand, the function l/x² has for x = 0 an infinite discontinuity at which its value becomes positive from both sides (Figs. 6 and 12). The function y = 1/(x² - 1), shown in Fig. 22, has infinite discontinuities both at x = 1 and at x = -1.
Finally, we shall illustrate by an example another type of discontinuity in which there does not exist a limit from the right or from the left. Consider the function
defined for all non-zero values of x. This function takes all values between -1 and + 1 as the number 1/x ranges through the values from (2n - 1/2)p to (2n+1/2)p, no matter what value has n. At the points x=2/(4n-1)p, the function will have the value -1, at the points x=2/(4n+1)p the value +1. We see from this that the function swings backwards and forwards more rapidly between the values +1 and -1 as x approaches closer and closer to the point 0 and that there occur in the immediate neighbourhood of the point x = 0 an infinite number of oscillations (Fig. 23 above).
It is interesting to observe that, in contrast
to the above example, the function y=xsin x/x
(Fig. 24 below) remains continuous at the point x = 0,
if we assign to it the value 0 at that point. This continuity is
due to the fact that, as the origin is approached, the factor x
damps the oscillations of the sine. Yet, in the neighbourhood of
the origin, the function y = x sin 1/x
does not change a finite number of times from monotonic increasing to monotonic
decreasing. On the contrary, it oscillates 
backwards and forwards an infinite number of times, the
magnitude of these oscillations becoming as small as we please as
the origin is approached. This example shows us that even the
simple idea of continuity admits all sorts of remarkable
possibilities foreign to our naive intuition.
There is one important fact which must be taken into account, if we are to give our ideas greater precision. It may happen that at a certain point a function is not defined by the original law, as for example at the point x = 0 in the last two examples discussed. We have then the right to extend the definition of the function by assigning to it any desired value at such a point. However, in the last example, we can extend the definition in such a way that the function also remains continuous at that point, namely, by setting y = 0 when x = 0. This can be done whenever both the limits from the left and from the right exist and are equal to each other; we need then only make the value of the function at the point in question equal to these limits, in order to make the function continuous there. In the case of the function y=sin1/a, this cannot be done.
1.8.3 Theorems on Continuous Functions: In conclusion, we quote the following important, general theorems, the proofs of which follow immediately from the remarks on operations with limits (cf. 1.6.4):
The sum, difference, and product of two continuous functions are themselves continuous. The quotient of two continuous functions is continuous at every point at which the denominator does not vanish.
In particular, it follows that all polynomials and all rational functions are continuous except at the points where the denominator vanishes. The fact that the other elementary functions, such as the trigonometric functions, are continuous will follow naturally from later considerations (2.3.5).
Exercises 1.7:
1. Prove that
2.Prove that
3. (a) Let f(x) be defined by the
equation y = 6x. Find a d,
depending on x, so small that
|f(x)-f(x)|
< e whenever |x -
x| < d,
where (1) e = 1/10; (2) e = 1/100 (3)
e = 1/1,000.
Do the same for
4. (a) Let f(x) = 6x in the interval 0 £ x £ 10. Find a d so small that |f(x1) - f(x2)| < e whenever |x1 - x2| < d, where (1) e = 1/100, (2) e is arbitrary, but > 0.
Do the same for
5. Determine which of the following functions are continuous. For those which are discontinuous, find the points of discontinuity:
