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There is probably no other instance in human intellectual history in which so much time and effort was spent merely to reach a satisfactory definition as that for limit. The concept is very closely related with two other fundamental concepts of mathematics and exists alongside both infinity and continuity. The Greek scholars were the first who seriously considered problems of continuity and infinity based on the concept of ‘number’.

Various attempts were made by them to include the concept of number into geometry. Being able to construct a line segment of any rational m/n length (m,n, the Greeks discovered around 400 B.C. that a diagonal of a unit triangle is an irrational number, which falls out of the number concept. Questions of continuity and infinity seem to have represented a complete mystery to the Greek scholars. The difficulties were clearly indicated by the famous Paradoxes of Zeno, of which it is worth quoting the following:

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What is the fallacy of the paradox of Achilles and the tortoise? Using concepts established some 2,500 years after Zeno, here is the explanation why Achilles can finally catch up to and pass the tortoise: Although the number of time intervals is infinite, the total amount of time is not necessarily infinite. Suppose the tortoise is given a head start of 3 meters and advances at the speed of 3m/s and Achilles ambles along at 6 m/s, Achilles will catch up to the tortoise at the end of (see Appendix 1).

The space above introduced by ‘…’ stands for the infinite number of decreasing fractions called a sequence, that add up to 1 second. Although it seems not so difficult to express every term of the sum with respect to its place – no matter how far it is- the entire process of computation of the sum of the infinite number of terms is not clear. On the other hand, even if one could intuitively estimate, that the terms of the above sequence tend to zero (converge to zero), as one chooses a term far enough, it is still unclear, how to justify one’s intuitive guess, because he/she might have intuition different from others.

The space ‘…’ in (1) reflects the distance between first attempts of ancient scholars to halt ‘leak’ of information about incompleteness of rational numbers they discovered, and the modern era development of old concepts such as differential and integral calculus, which solved and interpreted ancient and modern enigmas. Amongst the earliest and most significant contributors to rigor in calculus was A. Cauchy. He explained the meaning of the above expression ‘tends to zero as the term is far enough’ in following terms: ‘ When the successive values attributed to a variable approach infinitely a fixed value so as to end by differing from it by as little as one wishes, this fact is called the limit of the others.'(2, p. x)

This definition seems excessively vague from our viewpoint: the phrases ‘successive values’, ‘approach indefinitely’, ‘as little as one wishes’ are suggestive rather then mathematically precise. Therefore A. Cauchy’s definition needed to be refined in terms of formal mathematical language and this was done by H.E. Heine forty three years after the first publication of the above definition of limit. Heine defined the limit of a function f(x) at x0: ‘If given any , there is an such that for the difference is less in absolute value than , then L is the limit of f(x) for x=x0.’

This statement, which is now the accepted definition of limit, is absolutely unambiguous. With minor modifications, it applies to many other kinds of limiting processes, including sequences and series of numbers and functions, functions of several variables, complex functions etc. The paradox of Zeno regarding motion disappears once the definition of continuity based on Heine’s definition of limit is understood.

This also led to clear definitions of number, continuity, and derivative enabling nineteenth century scientists to provide a logically precise development of calculus. With an instrument as powerful as calculus, modern mathematicians solved problem of estimating volumes of solids formulated by Archimedes.. We strongly believe that it is impossible to teach someone this concept. However those lucky to touch it may feel as great a pleasure as those who understand the Bach harmonies.

In this essay we discuss only three applications of the concept, namely, the limit of sequence, the limit of series and the limit of function of one real variable supporting our reasoning with some samples. As the amount of words of the essay is strictly LIMITED we are not discussing a concept of infinity or continuity, which are based on the concept of limit. However when necessary we give outline of the former without further contemplations or speculation.

Limit of Sequence is a function defined on a set of real numbers for all positive integers. The intuitive definition of the sequence is already outlined in the Introduction. Roughly speaking, Definition 1:has a limit L if (xn-L) becomes arbitrary small for all sufficiently large values of n. In this case we write (2) . From this crude description, we would expect that the sequence 1,1,… has the limit 1, whereas the sequence has a limit 0 while the sequence 1,-2,3,-4,… does not have a limit.

On the other hand, our intuition is not sharp enough to deduct, whether the sequence has a limit and to compute the limit if there is one. Even a relatively simple sequence as , -1,1, -1,…, so-called oscillated sequence, could lead to the wrong intuitive conjecture, that it has two limits. So, we need an accurate definition of the ‘limit of a sequence’ based on which we would become capable to predict for any given sequences the existence of its limit and to evaluate it.

We emphasise that limit L should be a real number. Formally, Definition 1 means, that Definition 2: (3) . We can interpret it as follows: the proof that L is the limit of a given sequence , consists upon being given an of finding a value of N, such that the inequality holds for all values of n except at most a finite number, namely . The value of N will, in general, depend on the value of . Figure 1 illustrates Definition 2. All of the xn, except at most a finite number of terms, must be inside the parentheses.

Example 1: Consider the sequence , which can be expressed as xn=1/n (n=1,2,3,…). We already made a conjuncture that . Let us prove it. Following Definition 2 for given we must find N so that for all n;N: . Substitution of xn=1/n into the last inequality leads to , or, considering n;0 (4) . Thus if we choose N so, that 1/N;, then certainly (4) will hold since for all n larger than N. Now iff . Hence if we take any integer N such that , then (3) will hold for the given sequence with L=0. This proves thatalthough not one term of the sequence is equal to zero. Example 2 Examining the sequence xn=1(n=1,2,3,…) in terms of Definition 2 we can prove that our guess was correct. Really, if L=1 then for any .

So in this case (3) always holds and N does not depend on. Example 3. Consider the sequence 1,2,3,…, which can be expressed as .It can be proved by contradiction using the same -N method, which leads to the statement that the sequence 1,2,3,… has no limit or, that tends to infinity when n tends to infinity or diverges to infinity. Infinity is certainly not a number. Moreover, it is not a quantative concept. It is a quality of increase beyond bound.

Although the concept of infinity is difficult to grasp we can define it as not finite, contrary to finite, which is completely determinable by counting or measurement. Following the Galileo statement that there are as many squares as there are natural numbers, G. Cantor proved that the set of all integers, the set of all natural numbers, the set of all rational numbers and the set of all algebraic numbers are equivalent to the set of all natural numbers as they can be put in one-to-one correspondence with the infinitude of natural numbers. Following this concept we may think of all divergent to infinity sequences as having the ‘same manyness of elements’ as, by the definition, each sequence has one- to one correspondence with the set of all positive integers, hence belong to ‘aleph null’ set of infinity (1p.258-264).

Remembering though, that infinity cannot be expressed by any number (other things, like motherhood, happiness, faith belong to the qualitative category of concepts, that humans were hopelessly trying to describe by quantity), we discourage the idea to resolve indeterminate problems of substituting each term of the former by equal numbers in case of; for example, {} simply because there are no numbers equal to infinity. Coming back to Example 3 whatever large number we choose there are always terms, which would exceed it. This reasoning seems to breach Definition 2. Really, we determine infinite as not a number, hence in this case the inequality makes no sense.

So Definition 2 needs amendments such that there would not be a need to use the ‘suspicious’ concept in notation. We have already proved, that a sequence may have a limit, which is different to any of its terms, so the fact that ”is not a number should not contradict the perception of a limit. Following experience with the sequence=n we know, that Definition 3 For any given positive number M, there is an integer N (depending on M) such, that xn;M for every . This definition binds concepts of convergence to a finite limit as to a real number and divergence to infinity.

Is a sequence whose terms get ‘ too big’ as in Example 3, the only kind, which does not have a limit? We already consider a ‘suspicious’ example Let us suppose that it has a limit, so . Definition 2 has to be satisfied for any . Let us choose . Following (3) there would be a positive integer N such as . If n is even then the last expression means:/1-L/; 1/2 and for n odd it is /-1-L/;1. This implies that L should be less than half unit from 1 and less that half unit from -1, which is impossible. So by contradiction we proved that the sequence 1,-1,1,-1,… has no limit though the terms of the sequence all have absolute value 1 and hence are not ‘too big’. It is worth noticing, that the initial guess, that there may be two limits of the above sequence is proved to be wrong.