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In the writing of a limit \[\lim_{x\to a} f(x)=L\]

we will say "\(f(x)\) tends to \(L\) when \(x\) tends to \(a\)." One of the difficulties is that each part of the sentence, when considered alone, is meaningless; \(x\) cannot tend to \(a\) alone. Similarly: "\(f(x)\) tends to \(L\)" in itself is meaningless. The concept must be grasped in its entirety or not understood at all.

Defining the tangent as limit of secants offers two difficulties: the problem of the limit and, in addition, the fact that secants rest on two points, but if the tangent "rests" on a single point, then it can have any orientation, such as justifies that

\[x\mapsto |x|\]

has no tangent at \(x= 0\).

Defining the derivative as the slope of the tangent leaves us with the difficult task of defining the tangent before the derivative. For the integral, a sum of thin slices approximates the area but the limit when slices tend to a thickness of zero seems to imply that the area is a sum of zeroes. These difficulties are well known to analysis teachers. Many teachers avoid this by resorting to "hand waving" definitions and proofs about "informally approaching" or by "arbitrary closeness". Mathematical rigour is then lost.

Another way to circumvent the difficulty is to change the approach and use a version of nonstandard analysis – this is the choice we have made. Many difficulties disappear or become more palatable when using the concept of ultrasmall numbers. In particular, the limit is defined in such a way that each part of the sentence has a meaning on its own – hence didactic and intellectual steps are smaller. Most mathematicians have an intuitive idea of infinitesimals. These mental representations are often used to explain the fundamental concepts before rigorous formalisations are given. Recent work by Karel Hrbacek, based on Yves Péraire’s research offers a new formalisation of these ideas, mathematically rigorous yet still reasonably close to intuitive ideas and with a lower level of technical complexity. We have adapted this work to Geneva high school level.

There have been several attempts to use nonstandard analysis for teaching, by Keisler, Stroyan or Robert for instance. These previous theories used different approaches but one limitation was common to all: if \(h\) is infinitely small (in a way clearly defined in each theory) the derivative of \(f:x\mapsto x^2\) was easy at \(x = 2\) but difficult at \(x= 2 +h\). The approach used here does not have this drawback but - we hope - retains the advantages of Keisler, Stroyan and Robert.