Geodesic
Calculus without the concept of limit
Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia, Tome 6 (2014) no. 1, pp. 19-40.

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There are two different approaches to nonstandard analysis: semantic(model-theoretic) and syntactic (axiomatic). Both of these approachesrequire some knowledge of mathematical logic. We present a method basedon an ultrapower construction which does not require any mathematical logicprerequisites. On the one hand, it is a complementary course to a standardcalculus course. On the other hand, since it relies on a different intuitivebackground, it provides an alternative approach. While in standard analysisan intuition of being close is represented by the notion of limit, in nonstandardanalysis it finds its expression in the relation is infinitely close. Asa result, while standard courses focus on the " − technique, we explorean algebra of infinitesimals. In this paper, we offer a proof of the theoremon the equivalency of limits and infinitesimals, showing that calculus can bedeveloped without the concept of limit.
Keywords: hyperreals, calculus 
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Błaszczyk, Piotr; Major, Joanna. Calculus without the concept of limit. Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia, Tome 6 (2014) no. 1, pp. 19-40. http://geodesic.mathdoc.fr/item/AUPCDM_2014_6_1_a1/