THREE STEPS FOR SOLVING PROBLEMS USING THE DEFINITION OF THE LIMIT OF A SEQUENCE

Abstract

Formulation of the problem. An important component of mathematical training is learners’ command of the conceptual apparatus of mathematical disciplines, which, in most higher education institutions, is taught under the umbrella title Higher Mathematics, in particular Mathematical Analysis. The object of study in classical mathematical analysis is the function and various functional dependencies; its subject matter is the properties of functions, and the main tool for investigating these properties is the limit process. For courses in mathematical analysis, the central notion is that of a limit. This is explained by the fact that fundamental concepts such as the limit of a function, continuity, the derivative, and various types of integrals are introduced in terms of limits. Therefore, success in mastering these courses largely depends on the extent to which learners have mastered the notion of a limit, which makes the development of an effective strategy for forming the concept of the limit of a sequence relevant, including in tasks that require the practical use of the formal definitions of a limit.

Materials and methods. The study employed an analysis of scientific and methodological literature and textbooks in higher mathematics and mathematical analysis; a systematization of domestic and international approaches to introducing the concept of a limit; and a generalization of the author’s experience in organizing practical classes and selecting exercises in which proofs are constructed without appealing to ready-made theorems, relying only on the definition.

Results. A strategy was developed to develop learners’ understanding of the limit of a sequence and to consolidate this understanding in practical classes. An “algorithmic” approach to solving problems that apply the concept of the limit of a sequence is substantiated in the form of three consecutive steps based on the ε–n₀ definition of the limit of a sequence. The algorithm reduces learners’ cognitive load at the beginning of the topic, helps separate the heuristic search for estimates from the formal completion of the proof, develops skills in controlling error, and supports awareness of the dependence n₀(ε). The algorithm serves as a basis for methodological recommendations for practical classes and strengthens skills in proving inequalities as analytical tools.

Conclusions. A feature of the proposed method for applying the concept of the limit of a sequence in practical classes is that learners can independently grasp the meaning and importance of each detail in the definition of the limit of a sequence. Further research should be directed toward algorithmizing the application of definitions of the limit of a real-valued function of one variable at a point.