Poster presented at the Fourth International Congress on Mathematical Education, Berkeley, 1980, with abstract appearing in Abstracts of short communications, page C5. Intuitive infinitesimals in the calculus David Tall Mathematics Education Research Centre University of Warwick COVENTRY, UK Intuitive infinitesimals Intuitive approaches to the notion of the limit of a function “ lim x →a f ( x ) = l ” or “f( x)→ l as x → a” are usually interpreted in a dynamic sense “f( x) gets close to l as x gets close to a.” The variable quantity x moves closer to a and causes the variable quantity f( x) to get closer to l . As an implied corollary resulting from the examples they do, students often believe (see [5]) that f ( x) can never actually reach l. For l = 0 this gives a pre-conceptual notion of “very small” or “infinitely small.” A limit is often interpreted as a never-ending process of getting close to the value of l rather than the value of l itself. For instance, using the Leibnizian notation, students with a background of intuitive limiting processes might interpret dx as lim δx →0 δ x . The latter is not thought of as zero, but the process of getting arbitrarily close to zero. In the same way lim n→∞ a n is usually regarded as a never-ending process, so that 0 ⋅ 999 K9 K is regarded as less than 1 because the process never gets there. The extent of these phenomena will depend on the experiences of the students. In a questionnaire for mathematics students arriving at Warwick University, they were asked whether they had met the notation dy dx = lim δx →0 δ y δ x and, if so, they were asked the meaning of the constituent parts δx, δy, dx, dy. Of course some would have been told that dx and dy have no meaning in themselves, only in the composite symbol dy dx , a few others may have been told that dx is any real number and dy = ′ f ( x ) dx . Out of 60 students completing the questionnaire I classified their interpretation of dy as follows: