LIMITS,
of 1Vhen the difference be tween 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 4 + . . . is a variable which, Iv making n sufficiently large, may be made to differ from I by less than any assign able quantity. The variable may be always less than, always greater than, or sometimes greater, and sometimes less than, its limit. The series 1 — . . . is an example of the last, since the limit is
while the sums to one, two, three, etc., terms are 1, etc. The symbol = is used to indicate that a variable approaches a constant as a limit ; e.g. i +-I . . 1. In algebra the limit of a given function f (x) when r a is defined thus: If fur any positive value e, however small a number a can be found such that for all values of x, satisfying the inequality I x — < a, the corresponding values of Rd.) satisfy the in equality I f(x) b 1
(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 runnher 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 g or In case an expression of the form Pr) c° — becomes or for a particular value (-0 a of the variable, the true value is easily found by a rule due to L'ilepital, which con sists in replacing [(x) and (r) by their de rivatives, and, if necessary again, by their second derivatives, and so on. For exceptions to this rule
rind its applications, consult the works mentioned under CALCTIA7S. See also FnAcTioxs.
The theory of limits is among the most im portant in mathematics, the rigor of modern analysis depending 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 Newton's theory of prime and ultimate ratios found in the Prinripia (1687).
The limits of the roots of an equation are the numbers above and below which it is impossible that the roots should exist. The approximate sohition of numerical higher equations consists in bringing the limits of each root as nearly together as possible.
For historical and theoretical discussion. con sult Pringsheim, in the Eneyklopadie der mathe matisehen Wissenschaften, VOL i. (Leipzig. 1898).
LnermA (Lat., from Gk. Xe?gga, !rimer!, remnant, from Xehrew, leipein, to leave). An interval in the musical system of the ancient Greeks which to-day is designated as a diatonic semitone (a—bb), as opposed to the chromatic semitone (a—az), which latter the Greeks called apolomc. In our modern system the diatonic semitone is larger than the chromatic, but with the Greeks it was the reverse. They determined the limma by subtracting two whole tones, each in the proportion of S : 9, from the perfect fourth (3 : 4) and thus established the ratio 243 : 256; whereas the apotome or chromatic semitone was fixed in the proportion of 2048 : 2187.