LIMITE. — The l. (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 is found, and on this side of which the whole of it is found” (Met., V, 17, 1022 a 4–5; trans. by A. Carlini, Bari 1928, p. 174).
This notion, therefore, is originally derived from extension, in which the surface, the line, and the point are respectively the l. of the solid, the surface, and the line. Subsequently, as Aristotle himself and s. Tommaso note in his commentary on the passage cited, the concept of l. is transferred to movement, action, and time, as their point of departure and point of arrival, or those of a determinate part of them; and, still more generally, it is applied to any non-infinite entity. In this sense the l. is nothing other than the negation of further perfection and, according to the Thomistic thesis, has its principle in the limiting potency of the act (s. Tommaso, In V Met., lect. 19).
Aristotle also notes another meaning of l., 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); indeed, s. Tommaso explains in the commentary cited, “incipit cognitio rei ab aliquibus signis exterioribus quibus pervenitur ad cognoscendam rei definitio item; quo cum perventum fuerit, habetur perfecta cognitio de re”. Kant, by contrast, calls the concept of the noumenon a concept-l. (Grenze-griff), insofar as it limits the objective validity of sensible intuition, thereby indicating that such knowledge cannot extend its domain to what the intellect thinks. Nevertheless, it is not possible to become positively aware of the possibility of such a noumenon, but only problematically, because we lack every possibility of intuition other than sensible intuition; and therefore, for us, the territory beyond the sphere of phenomena is empty. “The concept of the noumenon is therefore only a concept-l. for circumscribing the claims of sensibility, and is consequently purely negative in its use” (Critica della ragion pura, Analitica dei principi, III; trans. by G. Gentile and G. Lombardo-Radice, I, Bari 1924, p. 251). A similar notion is also found in Lange, Riehl, Ardigo, Hoffding, and others belonging to the positivist current; while others, such as Paulsen, Max Wundt, Heidegger, and Carabellese, attribute to the concept of the noumenon a more positive meaning as well.
In mathematical analysis the notion of l. has important applications. A numerical set or a function is said to have an l., or lower or upper bound, when it is possible to determine a value l such that all the values of the set or function are not less than, or respectively not greater than, l. If a function has a minimum or a maximum in a certain interval, these coincide with the lower or upper l.; there is, however, also the case in which a set or a function has an extreme l. without the set having a first or last term, or without the function actually attaining a minimum or maximum value, since at the extreme the definition of an element of the set or of a value of the function no longer applies; for example, the set of proper fractions has no last element, although it has the unit as its upper l.; the function \(\frac{\sen x}{x}\), which is not defined at the point x = 0, nevertheless has at that point the upper l. 1. In such cases the idea of tendency toward the l. is applied, an idea that historically goes back to the method of exhaustion of Eudoxus and the Greek geometers and that, with Cavalieri and Wallis, provided the logical foundation of infinitesimal calculus, supplementing 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, given a positive number ε as small as desired, it is possible to 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. With suitable formal adaptations, this definition also serves to specify the meaning of tending toward infinity (positive or negative) and of tending toward a finite or infinite l., as x tends toward infinity or toward zero.