Calculus without the concept of limit

Abstract

There are two different approaches to nonstandard analysis: semantic (model-theoretic) and syntactic (axiomatic). Both of these approaches require some knowledge of mathematical logic. We present a method based on an ultrapower construction which does not require any mathematical logic prerequisites. On the one hand, it is a complementary course to a standard calculus course. On the other hand, since it relies on a different intuitive background, it provides an alternative approach. While in standard analysis an intuition of being close is represented by the notion of limit, in nonstandard analysis it finds its expression in the relation is infinitely close. As a result, while standard courses focus on the ε − δ technique, we explore an algebra of infinitesimals. In this paper, we offer a proof of the theorem on the equivalency of limits and infinitesimals, showing that calculus can be developed without the concept of limit.