On introduction of a concept of limit in mathematics courses

Dovbysh S.

AESC of M.V.Lomonosov Moscow State University, Russia, 121357, Moscow, Kremenchugskaya, 11. Phone +7 (926) 1232836, e-mail:

The fundamental concept of a limit (of sequence and function) turns out to be difficult to study and many students have problems with its assimilation. We found some typical imperfections in presentation of this concept in many courses, as well as specific approaches and features of some courses designed to simplify its study. The main disadvantages are as following:

1) Definitions are given formally, without explaining their essence at a visual level; in a significant part of textbooks there are no graphic illustrations. It seems useful to illustrate two or three concrete examples of calculating the limit, using only its intuitive concept and visual reasoning (as all mathematicians did before the introduction of strict definitions). This approach was presented in old textbooks, but now it has been largely banished, although it sometimes occurs, especially in textbooks for universities with a small mathematics program.

2) Since the essence of the concept of limit has not been clarified, the relationship between the definitions of the limit of a function for different cases of the limit value (finite number or infinity with or without a sign) and different cases of the variable tending remains unclear. Many textbooks do not clearly explain what the general scheme of constructing the limit is.

3) In fact, the concept of the limit of a function is based on the concept of «a variable tending to the limit» in relation to both values of the function and the independent variable. As it turned out, most of the old classical textbooks discuss this concept, first on an intuitive level, and then in stricter terms for each specific case of tending. The definition of the limit of a function turns out to be derived from the concept of the limit of a variable. This approach contributes better to understanding the essence of the concept of limit, and also follows its origin and historical development. But now it is almost completely banished from modern textbooks. A.Cauchy himself used the concept of a limit precisely in terms of an unclassified variable. N.N. Luzin, supporting the described approach, definitely spoke in favor of the «widest introduction», for pedagogical purposes, of «changes of variables» and at the same time he rejected the «reproach of laxity». He warmly supported M.Ya.Vygodsky's textbook «The foundations of the infinitesimal calculus» (1931, 1933), consistently written in this style, and with its detailed motivation.

Let us note in this connection that a number of other authoritative authors (A.Poincare, F.Klein, H.Lebesgue, A.N.Kolmogorov and especially M.Ya.Vygodsky) spoke about the need for an approach in teaching mathematics (especially for non-mathematicians) based on the use of clarity and intuition, and not pure logic at all, with subsequent «logical refinement and purification» (M.Ya.Vygodsky) of ideas and non-strict concepts that follows the historical path of the development of a science.


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