Local (L, ε)-approximation of a function of single variable: An alternative way to define limit

At the start of a freshman calculus course, many students conceive the classical definition of limit as the most problematic part of calculus. They not only find it difficult to understand, but also consider it of no use while solving most of the limit problems and therefore, skip it. This paper reformulates the rigorous definition of limit, which may be looked upon as a local approximation of a function by a zero degree polynomial. For this purpose a notion of local (L, ε)-approximation is introduced. The approach conforms with all theoretical aspects of limit and continuity. Computational procedures and use of software for the solution of limit problems where necessary are discussed. It is expected that the suggested approach will be easy to follow by freshman calculus students.