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Back to Classics: Teaching Limits Through Infinitesimals

The usual $ε,δ$-definition of the limit of a function (whether presented at a rigorous or an intuitive level) requires a "candidate $L$" for the limit value. Thus, we have to start our first calculus course with "guessing" instead of "calculating". In this paper we criticize the method of using calculators for the purpose of selecting candidates for $L$. We suggest an alternative: a working formula for calculating the limit value L of a real function in terms of infinitesimals. Our formula, if considered as a definition of limit, is equivalent to the usual $ε,δ$-definition but does not involve a candidate $L$ for the limit value. As a result, the Calculus becomes to "calculate" again as it was originally designed to do.