The aim of this paper is to examine students' understanding of the limiting behavior of a function from [set of real numbers][superscript 2] to [set of real numbers] 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. (Contains 5 figures.)