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This paper describes a research project implemented in a course of Real Analysis II. The aim of this project is to analyse students' comprehension of the limiting behaviour of a function from "R"[superscript 2] to "R" at a point "P"[subscript 0] focusing on the formal definitions. Alternative definitions and other statements are discussed, some of which meet both sufficient and necessary conditions, while others meet only sufficient conditions; some involve neighbourhoods and others 'paths'. Emphasis is placed on the criteria according to which students choose one of the given definitions, equivalence or not of these and their overall influence on the concept images formed by the students for the limit concept. In particular, the paper discusses how students come to a conclusion about the existence of the limit of a function in "R"[superscript 2] through the consideration of contour curves. We also present and analyse data from a written test as well as from the conducted interviews.

Abstract: The aim of this paper is to examine students’ understanding of the limiting behavior of a function from to at a point P This understanding depends on which definition is used for a limit. Several definitions are considered; two of these concern the notion of a neighborhood of P, while another two are directed at the consistency of limits obtained by restricting the function to lines or half-lines passing through P. A case study is presented involving four university students studying Mathematics. Comments are made about their abilities in working with each definition, associated images that were evoked, and how they related one definition to another (including the issue of logical equivalence). The influence of the students’ previous experience in handling limits for real functions of one variable is also discussed. [Copyright &y& Elsevier]