Is the Cauchy Definition of Limit Limited in Rigor? A Foundational Revisit

The limit definition, or the ϵ-δ definition, as it has come down to us through two centuries, is still beset by suspicion from critics, being questioned for its level of rigor. The issue seems to stem from the precision of its statement and the logical soundness of its expression. There have been some questions on the soundness of its arguments in the midst of infinity and have called for a more ‘discrete’ approach. There have been arguments which say that the formal definition is invalid because ‘infinity does not exist’. The aim of this paper is to show that, at least within the framework of mathematical analysis and the tenets of mathematical logic, the ϵ-δ definition is logically sound, and its level of precision has not been eroded by years of practice and advancement in the field, and in fact still serves as a spring board for further analytical studies.