\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2012c.00580}
\itemau{Range, R.Michael}
\itemti{Where are limits needed in calculus?}
\itemso{Am. Math. Mon. 118, No. 5, 404-417 (2011).}
\itemab
Higher mathematics is characterized by the use of limits and the author states: ``As experienced mathematicians, we know that limits ultimately cannot be avoided" (p. 404). This paper, however, investigates the question whether the concept of limit or the use of infinitesimal quantities were actually needed in the development of the calculus. To be sure, history is actual and factual, i.e., a later logical interpretation cannot be an equivalent description of what happened in mathematics. Although the author uses historical facts, he did not write a historical paper: nevertheless he offers an interesting article. By the way, Descartes in constructing normal lines (perpendicular to tangents) did not work with secants becoming tangent lines, but he used osculating circles (i.e., he worked with a higher order of contact). Of course, for functions $f(x)$ of the form $f(x) = (x-a) q(x)$, the derivative $f'(a)$ can easily be calculated as $q(a)$ by simple algebraic operations alone (although the assumption `$q(x)$ is continuous in $a$' is not trivial and demands also a limit process, though the author accepts the obvious intuition of the students). Moreover, each function with an expansion into a power series at the point $a$ (analytical functions in the sense of Euler) allows the mentioned easy calculation of derivation. However, here the limit process is clearly embedded in the possibility of an expansion of a function in a series -- otherwise there is no bridge between power series and any function not given as a power series (let us take trigonometric functions as an example). It was Lagrange who tried to pull forward the limit process by starting with power series. There is an interesting paper by Juschkewitsch on this attitude compared with Euler's conception [{\it A. P. Juschkewitsch}, ``Euler und Lagrange \"uber die Grundlagen der Analysis'', Sammelband Leonhard Euler, Dtsch. Akad. Wiss. Berlin 224--244 (1959; Zbl 0107.24701)]. For all that, let us assume we have no problems with analytical functions. But we cannot agree with the author's statement: ``This simple approach to derivation is not limited to algebraic functions, but applies just as easily to formal power series, that is to the most general type of functions accepted in the 17th and 18th centuries" (p. 405) or ``The real need for limits begins with integrals" (p. 405). A typical and important counterexample is the initial state of solutions of differential equations, among which the mathematicians of the 18th century found curves (functions) which cannot be represented by any power series, but by drawing them freely by hand. Obviously, here we need transcendental processes instead of algebraic operations. In opposition to the author, from a methodological viewpoint one can take this optimistic attitude that the easy algebraic operations for a broad class of functions as intuitive examples do explain the method for ``confusing" transcendental functions. But all in all, this paper is encouraging for all those who teach this subject mathematically.
\itemrv{R\"udiger Thiele (Halle)}
\itemcc{I45 I49}
\itemut{derivative; infinitesimal quantities; limit}
\itemli{doi:10.4169/amer.math.monthly.118.05.404}
\end