Limit

LIMIT. - The limit (Greek πέρας, Latin terminus, German Grenze) is defined by Aristotle as "the extreme point of a thing: that first point, that is, beyond which nothing of it is found, and on this side of which everything of it is" (Met., V, 17, 1022 a 4-5; trans. by A. Carlini, Bari 1928, p. 174).

This notion is therefore originally derived from extension, in which surface, line and point are the limits respectively of volume, surface and line. Subsequently, as Aristotle himself and St. Thomas note in his commentary on the passage cited, the concept of limit is transferred to movement, action, time, as the point of departure and the point of arrival of these or of a determinate part of them, and even more generically it is applied to any entity that is not infinite. In this sense the limit is nothing but the negation of further perfection and, according to the Thomistic thesis, has its principle in the limitative potency of act (St. Thomas, In V Met., lect. 19).

Aristotle also notes another sense of limit, namely "the essence, the quiddity of each thing: for this is the term of cognition; and if of cognition, also of the thing" (Met., loc. cit., 9-10); in fact, St. Thomas explains in the commentary cited, "incipit cognitio rei ab aliquibus signis exterioribus quibus pervenitur ad cognoscendam rei definitionem; quo cum perventum fuerit, habetur perfecta cognitio de re." On the contrary, Kant calls the concept of the noumenon a limit-concept (Grenzbegriff), insofar as it limits the objective validity of sensible intuition, thus indicating that such knowledge cannot extend its domain to what the intellect thinks. Nevertheless, the possibility of such a noumenon cannot be positively accounted for, but only problematically, because we lack any possibility of intuition other than the sensible; and therefore the territory beyond the sphere of phenomena is for us empty. "The concept of the noumenon is therefore only a limit-concept for circumscribing the pretensions of sensibility and is therefore of purely negative use" (Critique of Pure Reason, Analytic of Principles, III; trans. by G. Gentile and G. Lombardo-Radice, I, Bari 1924, p. 251). A similar notion is also found in Lange, Riehl, Ardigò, Høffding, and others of the positivist current; while others, such as Paulsen, Max Wundt, Heidegger, Carabellese, attribute to the concept of the noumenon also a more positive sense.

In mathematical analysis the notion of limit has important applications. A numerical set or a function is said to have a limit or bound, lower or upper, when it is possible to determine a value l such that all the values of the set or function are not less, or respectively not greater, than l. If a function in a certain interval has a minimum or a maximum, these coincide with the lower or upper limit; it also happens, however, that a set or a function has an extreme limit, without the set having a first or last term, or without the function actually assuming a minimum or maximum value, inasmuch as at the extreme the definition of an element of the set or of the value of the function no longer holds; e.g., the set of proper fractions has no last element, although it has unity as its upper limit; the function frac{sin x}{x}, not definable at the point x = 0, has there the upper limit 1. In such cases one applies the idea of tendency to the limit, which historically goes back to the method of exhaustion of Eudoxus and the Greek geometers and which with Cavalieri and Wallis offered the logical foundation of infinitesimal calculus, integrating the Aristotelian concept of potential infinity: l is said to be the limit of the function y of x, as x tends to a, if, a positive number ε chosen as small as one pleases, one can determine a neighborhood σ of a such that, for every point of σ distinct from a, the corresponding value of y differs from l, in absolute value, by less than ε: that is, lim. y(x) = l, if y - l < ε for x - a < σ. This definition, with appropriate formal adaptations, serves also to specify the meaning of tendency to infinity (positive or negative) and of tendency to a finite or infinite limit, as x tends to infinity or to zero.

Filippo Selvaggi