Many researchers (Artigue, 2000; Bezuidenhout, 2001; Cornu, 1991; Dorier, 1995) have noted the vital role limit plays as a foundational concept in analysis. The vast majority of topics encountered in calculus and undergraduate analysis are built upon understanding the concept of limit and being able to work flexibly with its formal definition (Bezuidenhout, 2001). The purpose of this study was to: (1) Develop insight into students' reasoning about limit in relation to their engagement in instruction designed to support their reinventing the formal definition of limit, and; (2) Inform the design of principled instruction that might support students' attempts to reinvent the formal definition of limit. The first objective was at the foreground of the study and was set against the broader background goal of contributing to an epistemological analysis (Thompson & Saldanha, 2000) of the concept of limit of a real-valued function and its formal definition. A central aim of epistemological analysis is to identify and understand key aspects of what might be entailed in coming to understand a particular concept in relation to engagement with appropriate instruction.

In separate teaching experiments, two pairs of students successfully reinvented a definition of limit capturing the intended meaning of the conventional ϵ-δ definition. Analyses of the data generated in the teaching experiments revealed thematic elements of students' reasoning in the context of reinvention. For instance, the students' ability to shift from an x-first perspective (i.e., focusing first on x-values approaching the limiting value a and then on corresponding y -values approaching a particular value L) to a y-first perspective (i.e., considering first a range of output values around a predetermined limit candidate L and then establishing the existence of an interval of input values that would result in corresponding output values within the specified range) appeared paramount in their attempts to reinvent and reason coherently about the formal definition. This dissertation traces the evolution of the students' definitions over the course of two ten-week teaching experiments, and highlights thematic findings which point to what might be entailed in coming to reason flexibly and coherently about limit and its formal definition.