The subject of this paper occupies a strange position among the fundamental concepts of mathematics. The beginner uses limits with equal freedom, or, as some would say, with equal recklessness, in analysis and geometry, but whereas the time comes when the analytical limit is submitted to the severest scrutiny, the examination of the geometrical limit forms no part of any university course and comes into no treatise. There is not even an essay on the subject in the collection of Questioni riguardanti la Geometria elementare, edited by Enriques, or a section dealing with it in Klein’s Elementarmathematik vom höhere Standpunkte aus. When his turn comes to teach, every mathematician knows all about limits in analysis, is familiar with half a dozen ways of looking at them, and indeed has a basis of thorough understanding from which to exercise his discretion as to the treatment proper for one class of students or another. Of limits in geometry, except perhaps in three special cases, he has probably never heard a critical word.