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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 R2 to R at a point 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 R2 through the consideration of contour curves. We also present and analyse data from a written test as well as from the conducted interviews. [ABSTRACT FROM AUTHOR]