A perfect geometric illustration is found in the sequences I and C of inscribed and
scribed regular polygons of the circle. Here
every C is > every I; also C„ I„
Algebraically, if C,, C., C. . . . Cos+ .. be the sequence (0) of odd convergents and
C., . . . Cos . . . the sequence (E) of even con vergents in an interminate continued fraction 1 1 1 as — — — .... then every Cos+, 1+1+1+1+6+1+ >Can , also Can +i—C2(ft +k)+1
The difference IV—L( is a variable small at will and is called infinitesimal (a); its limit is O. The quotient of two a's will generally be a variable; if it has a finite limit L, the a's are named of the same order; if the limit of the quotient is 0 (or ea ), then the numerator (or denominator) is of higher order. If any a be chosen as standard, it is called principal infinitesimal; any other whose pth root is of the same order as the principal a is itself said to be of pth order.
Easy theorems are now proved as to the limits of the sum, difference, product, quotient or variables. In general: If R(s, v, w, ...) be a rational function of simultaneous variables, u, v, w, . . ., and if is, v, w . . . have limits, 1, m, n . . R(u, v, w.....) has a limit R(l, m, n . . .)—always provided that this latter does not involve a division by 0, which has no sense.
If two V's differ at most by a a, and one has a limit, the other has the same limit. Herewith there becomes possible a Calculus of the Limits of Variables instead of the Variables them selves. These limits are often far the more important, as we shall soon see.
A variable V (or sequence . . .) is bounded above when we may assign a value M that it cannot exceed; then there is a certain smallest number its upper limit which it can not exceed. Similarly, it is bounded below when we may assign an m below which it cannot sink; then there is a certain greatest number, its lower limit, tinder which it cannot descend. If V may assume either of these limits as one of its values, then that limit is attainable and is a Maximum or a Minimum; otherwise it is unattainable. If V be a proper
fraction, its limits, 0 and 1, are not attainable.. When V may assume every value between its attainable limits, a and b, it is said to vary continuously in the interval [a, bj. But if a, or b, or both be unattainable, we shall say that it is continuous in [a-I-0, Id or [a, b--01, or [a+0, b-0].
When to values of one magnitude correspond values of another, the magnitudes are called Functions of each other (Leibnitz). The one to which arbitrary values may be supposed given is called the argument or independent variable; the other, whose corresponding values may be reckoned or observed (or which at least exist), is called the function. Such are a number and its logarithm or sine; the radius of a sphere and its surface or volume; the elasticity of a medium and the velocity of an undulation through it; etc. . . . The general functional connection of x and y is expressed by F(x, y)=0. If this F be an entire poly nomial in x and also in y, the F is algebraic, otherwise transcendental. If F be solved as to y, thus y=f(x), then y is an explicit function of x; otherwise, an implicit function. If t(x) be the quotient of two entire polynomials in x, then f(x) is a rational function of x; otherwise, irrational. If to any one value of x there corresponds only one value of y, then y is a one-valued or unique function of x; if x be also a unique function of y, then there exists between x and y a one-to-one correspondence.
If y = f(x) and (y) express the same correspondence between x and y, then f and denote inverse functions. A function may re duce to a constant; as xa 1, for every finite x when n=0.
As x ranges in [a, h], f(x) will also range. Similarly f(x) may have an upper limit M and a lower limit m; then f(x) is bounded in [a, b], [in, 31] is its interval and M —m its oscillation. If either m or M be absent (or= ), this oscil lation is co. If we cut [a, b] into n sub-inter vals (at, bk) (k=1, . . . ii), then plainly the upper limit of f(x) will be M in at least one lak, bk] and > M in none; the oscillation will not be > M —m in any [ak, If as x approaches c, no matter how, f(x) approaches f(c) as its limit, then f(x) is con tinuous at c (i.e. for x= c). Or, if f (x) be bounded in [c —a, c + a] and if the limit of its oscillation be 0 for a vanishing, then f(x) is continuous at c. That is, we must be able to make and keep the oscillation of f(x) small at will by making and keeping the fluctuation in r small at will.