LIMITS, TrrEonv OF, The importance of the notion of a limit in mathematics can not be over-estimated, as many branches of the science, including the differential +calculus and its adjuncts, consist of nothing else than tracing the consequences which flow from this notion. The following are simple illustrations of the idea: The sum of the series 1 + + etc., approaches nearer and nearer to 2 as the number of terms is increased; thus, the several sums are 11, lt, 1-R, etc., each sum always differing •from 2 by a fraction equal to the last of the terms which have been added; and since each K:lenominator is double of the preceding one, the further the series is extended, the less the difference between its sum and 2 becomes; also this difference may be made smaller than any assignable quantity—say, riruico,3,—by merely extending the series till the last 'denominator becomes greater than 100,000 (for this, we need only take 18 terms; 3 terms more will give a difference less than noThos; and so on); again, the sum of the series can never be greater than 2, for the difference, though steadily diminishing, still sub -sists; under these circumstances, 2 is said to be the limit of the sum of the series. We :see, then, that the criteria of a limit are, that the series, when extended, shall approach nearer and nearer to it in value, and so that the difference can be made as small as we please. Again, the area of a circle is greater than that of an inscribed hexagon, and less than that of a circumscribed hexagon; but if these polygons be converted into figures of 12 .sides, the area of the interior one will be increased, and that of the exterior dimin ished, the area of the circle always continuing intermediate in position and value; and :as the number of sides is increased, each polygon approaches nearer and nearer to the oircle in size; and as, when the sides are equal, this difference can be..made as small as
we please, the circle is said to be the limit of an equilateral polygon the number of whose sides is increased indefinitely; or, in another form of words commonly used " the polygon approaches the circle as its limit, when its sides increase without limit," or Agrain, " when the number of sides is infinite, the polygon becomes a circle." -When we use the terms " infinite " and " zero " in mathematics, nothing more is meant than that the quantity to which the term is applied is inereasing without limit or diminish ing indefinitely; and if this were kept in mind there would be much less confusion in the ideas connected with these terms. From the same cause has arisen the discus sion concerning the possibility of what are called. vanishing fractions (i.e., fractions Limma. 4,0 Lincoln.
whose numerator and denominator become zero simultaneously) having real values; x9 — 1 0 thus — when x = 1; but by division SVC find that the fraction is equal to x 1, — 1 —if which = 2, when x = 1. Now, this discussion could never have arisen had the question — 1 been interpreted rightly, as follows: approaches to 2 as its jimit, when x x — 1 tinually approaches 1 as its limit, a proposition which can be proved true by substituting successively 3, 2, 114-0-, etc., when the correspondino. values of the fraction are 4, 3, 2i, 2110-, 2th, etc. The doctrine of limits is emplOyed in the differential calculus (q. v.). The best and most complete illustrations of it are found in Newton's .Prineipia, and in the chapters on maxima and minima, curves, summation of series, and integration generally, iu the ordinary works on the calculus.