LIMITS. 278 lilMOGES. LIMITS, Tjieokv of. lien the differonce be- twi'L'ii a variable and a constant quantity may become and remain less in absolute value than any assignable quantity, however small, the con- stant is called the limit of the variable, e.g. the sum of two, three, four, . . . n terms of the series i + } + i + A+- • • >« ^ variable which, by making n sulliciently large, may be made to differ from 1 by less than any assign- able quantity. The variable may be always less tlian, always greater than, or sometimes greater, and sonietimes less than, its limit. The series 1 — J + 1 — i + • • • is an example of the last, since the limit is j, while the sums to one, two, three, etc., terms are {, J, f, etc. The symbol =^ is used to indicate that a variable approaches a constant as a limit; e.g. i + J; + 1 + . . . ^ 1. In algebra the limit of a given function f (x) when x =h » '^^ defined thus! If for any positive value c, however small a number o can be found such that for all values of j;, satisfying the inequality x — o <, a, the corrcsi)ouding values of f(x) satisfy the in- c(iualitv I fix) —b |<e, then 6 is called the limit of Hx). E.g. let fix) = ar", and suppose X:^0; select e as 0.001; then (1) J »— 0|<a, (2) I a;' — 0| < 0.0001, show that for o = 0.1, any value of x satisfying ( 1 ) gives an x' satis- fying (2), therefore, bv making e small enough, jjs j_ as a; = 0. (The notation x stands for the ahsoluie rulue of x.) The fundamental propositions of limits are: (1) H two variables are always equal or in a constant ratio as they approach their limits, their limits nrv. equal or are in the same ratio. (2) The limits of the algebraic sum of a finite number of variables is the sum of their limits. (3) The limit of the product of a finite number of variables is the product of their limits, all the quantities being finite. (4) The limit of the quotient of two variables is the quotient of the limits, if the expression does not assume the form ^ or f(^)
In case an expression of the form becomes ^ or -^ for a particular value x^a of the variable, the true value is easily found bv a rule due to I/Hojiital, which con- sists in" replacing fix) and ipix) by their de- rivatives, and. if necessary again, by their second derivatives, and so on. For exceptions to this rule and its applications, consult the works mentioned under C.VLcri.us. See also Fraction's. The theory of limits is among the most im- portant in "mathematics, the rigor of modern analysis depcndinj; upon the high state of per- fection of this theory. The mensuration of curves and surfaces, the treatment of series (q.v.), and the foundations of calculus (q.v.) rest upon it. The modern methods of limits is a development of Xewton's theory of prime and ultimate ratios found in the Princi-pia (1687). The limits of the rnntx of an equation are the numl)ers above and below which it is impossible that the roots should exist. The approximate solution of numerical higher equations consists in bringing the limits of each root as nearly together as possible. For historical and theoretical discussion, con- sult Pringshcim, in the Enci/klopadie der mathe- TiuiliKohen Wif!senJtchaften, vol. i. (Leipzig. 1808). LIM'MA (Lat., from Gk. XeiMA«t, teimma, remnant, from Xdveiv, leipein, to leave). An interval in the musical system of the ancient Greeks which to-day is designated as a diatonic semitone (a — bl)),"as opposed to the chromatic semitone (a — aff), which latter the Greeks called apolomc. In our modern system the diatonic semitone is larger than the chromatic, but with the (Jrceks it was the reverse. They determined the limma by subtracting two whole tones, each in the i)roportion of 8 : », from the perfect fourth (3 : 4) and thus established the ratio 243 : 250 ; whereas the apotome or chromatic semitone was fixed in the proportion of 2048 : 2187. LIMN.ffi'A. See Pond-Snail. LIMNANDER DE NIEUWENHOVE, Itm'- niui-dei- da ue'vcu-ho'vc, Ar.maxd ilAKlE Guis- I,AI.-, Baron (1814-92). A Belgian composer, born at Ghent. lie was educated at the Saint Acheul Jesuits' College, near Amiens, afterwards at Fribourg, and later studied composition with Fetis. Upon his return to Belgium from Switz- erland, he began the work of composition, and also conducted an amateur symphonic society at ilechlin. In connection with this lie founded a choral society, the Reunion Lyrique, and he wrote some fine choruses for unaccompanied male voices, besides a Ilcqiiicm Mass with or- gan; a Slahat Mater, with orchestra; a sonata for piano and violoncello; Lcs Druidcs, operatic scenes produced at the Paris Conservatory (1845); a symphony called ha fin des mois- nons; a Te Deum; and some cantatas. In 1847 he removed to Paris, where his comic opera Lcs ilontcncyrins was presented at the Opera Comique (1849), followed by Le chateau dc la baihc blcue (1851) and Vronne (18.59), while his Maitre-chanteur was given at the 0)fiy: (lS5:i) and again in Brussels (1870) under the title Miixiinilicn. LIMNAN'THEMUM. A genus of plants. See ViLLAij.siA. LIMNO'RIA. See Gribble. LIMOGES, le'mf)zh'. The capital of the De- partment of Haute-Vienne, France, and the ancient capital of Limousin (q.v.), situated on the right bank of the Vienne, 215 miles .south- southwest of Paris (Map: France, H 0). It is a very old town, with narrow, crooked streets in tlic older quarters, and many buildings and monu- ments attesting its antiquity. The newer por- tions of the town, built up after the demolition of the old fortificaticms. arc modern in appear- ance. The most notable building is the Cathedral of Saint Etienne, begun in 1273 on the site of an older church and completed only in the second half of the nineteenth century. The interior is decorated in Renaissance style and contains a delicately ornamented rood-loft of the sixteenth century." The town hall is a modern Renaissance building (1878-Sl) and has a collection of paintings and sculptures. Limoges has some Roman remains and a number of private houses dating from the Middle Ages. The educational institutions of the town consist of a lyc^e, a theological seminary. pre]>aratory schools of medicine and pharmacy, a national school of decorative arts, a library of about 30.000 vol- umes, a museum of ceramics, and a meteorologi- cal observatory. Limoges has since ancient times been known for its' artistic industries. In the Jliddle Ages it was celebrated for its mint and gold ware and later for its enamels. In 1730
