Original Research

Comparing the finite and infinite limits of sequences and functions: A mathematical and phenomenological analysis and its implications in Spanish textbooks

Mónica Arnal-Palacián, Francisco J. Claros-Mellado, María T. Sánchez-Compaña
Pythagoras | Vol 45, No 1 | a774 | DOI: https://doi.org/10.4102/pythagoras.v45i1.774 | © 2024 Mónica Arnal-Palacián, Francisco J. Claros-Mellado, María T. Sánchez-Compaña | This work is licensed under CC Attribution 4.0
Submitted: 27 July 2023 | Published: 21 May 2024

About the author(s)

Mónica Arnal-Palacián, Department of Mathematics, Faculty of Education, University of Zaragoza, Zaragoza, Spain
Francisco J. Claros-Mellado, Department of Mathematics, IES Calderón de la Barca, Pinto, Spain
María T. Sánchez-Compaña, Department of Didactics of Mathematics, Didactics of the Social Sciences and Experimental Sciences, Faculty of Education, University of Malaga, Malaga, Spain



The purpose of this article is to conduct a mathematical and phenomenological comparison of three concepts: (1) the finite limit of a function at a point, (2) the finite limit of a sequence, and (3) the infinite limit of a sequence. Additionally, we aim to analyse the presence of these concepts in Spanish textbooks. The methodology employed is exploratory and descriptive. Our mathematical comparison revealed differences in several areas, including the dependence between variables, the involved infinite processes, the types of infinity, the dimensioning for each variable, and the intuition of continuity in the interval. Our phenomenological comparison found a correspondence between phenomena using a formal approach, but differences in phenomena when using an intuitive approach. Finally, our analysis of textbooks revealed that all three limits are most commonly presented in the verbal representation system and definition format.

Contribution: This study has contributed to the teaching and learning of the notion of limit.


Finite limit; infinite limit; sequence; function; mathematics comparison; phenomenology; textbooks; intuitive approach.

Sustainable Development Goal

Goal 4: Quality education


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