Keyword: «limit point»

Limit of a function of a real variable is one of the first topics of calculus course students meet at the university lectures and exercise classes. For the first-year students, their first difficulties in math studying are connected with this topic; for example, the new language of mathematical signs and terms. Different textbooks explain this topic in different ways in spite of its being well known to professors and methodologists. Thus, the choice of the correct approach to the topic “Limit and continuity of a function” depending on the training area, skills, time limits and learning motivation of students is really important. The aim of the study: the article is devoted to the analysis and classification of the approaches to the definition of the function limit in a given point; they are different in the requirements to the point according to the domain of a function. The methodology of the study: the classification is obtained by reviewing of textbooks (theoretical method); we demonstrate how the choice of the approach to the definition of the function limit affects problem solving in the practical course (empiric method). Results of the study: the reviewing and comparing of the limit definitions given in different textbooks lead to a theoretical result, which is the classification of the definitions of the limit by the characteristic mentioned above; the analysis of the learning tasks examples on the limits calculation and checking of function continuity in the light of this classification provides practical application of our findings in the learning process. Theoretical value of the article: principles of classification of the limit definition conditions are given; the classification is constructed. Practical value of the results: using the information from the article, teachers can design lessons that help students understand the concepts of limits and continuity more deeply. Students can then apply this knowledge to solve problems related to these topics with greater confidence. Besides the methodological meaning for studying of just this specific topic, the achieved results are useful for the whole calculus studying to establish strictness and detailing adequate to the specialization and training level of the students. The reaction of the students to this initial topic, their questions, difficulties, and the more or less successful ways they overcome these difficulties will serve as a guide for the teacher in choosing a further strategy for working with students. This is the additional practical and methodological value of the obtained results.