The limit concept is a fundamental mathematical notion both for its practical applications and its importance as a prerequisite for later calculus topics. Past research suggests that limit conceptualizations promoted in introductory calculus are far removed from the formal epsilon-delta definition of limit. In this article, I provide an overview of research on limit conceptions and compare the competing theories for how the formal definition is constructed. The findings and assumptions of these research designs are discussed, leading to discussion of an alternative theory, corresponding pedagogical practices, and future research needs.