Relatively little is known about how students come to reason coherently about the formal definition of limit. While some have conjectured how students might think about limits formally, there is insufficient empirical evidence of students making sense of the conventional [superscript epsilon]-[delta] definition. This paper provides a detailed account of a teaching experiment designed to produce such empirical data. In a ten-week teaching experiment, two students, neither of whom had previously seen the conventional [superscript epsilon]-[delta] definition of limit, reinvented a formal definition of limit capturing the intended meaning of the conventional definition. This paper focuses on the evolution of the students' definition, and serves not only as an existence proof that students can reinvent a coherent definition of limit, but also as an illustration of "how" students might reason as they reinvent such a definition.