Page:EB1911 - Volume 11.djvu/316

Rh assigned by the rule of calculation real. In the most important cases the domain of the argument of a function of one variable is an interval, with the possible exception of isolated points.

6. Limits.—Let $$f(x)$$ be a function of a variable number $$x$$; and let $$a$$ be a point such that there are points of the domain of the argument $$x$$ in the neighbourhood of $$a$$ for any number $$h$$, however small. If there is a number $$L$$ which has the property that, after any positive number $$\epsilon$$, however small, has been specified, it is possible to find a positive number $$h$$, so that $$|L - f(x)| < \epsilon $$ < for all points $$x$$ of the domain (other than $$a$$) for which $$|x - a| < h $$, then $$L$$ is the “limit of $$f(x)$$ at the point $$a$$.” The condition for the existence of $$L$$ is that, after the positive number $$\epsilon$$ has been specified, it must be possible to find a positive number $$h$$, so that $$|f(x') - f(x)| < \epsilon $$ for all points $$x$$ and $$x'$$ of the domain (other than $$a$$) for which $$|x - a| < h$$ and $$|x' - a| < h$$.

It is a fundamental theorem that, when this condition is satisfied, there exists a perfectly definite number $$L$$ which is the limit of $$f(x)$$ at the point $$a$$ as defined above. The limit of $$f(x)$$at the point $$a$$ is denoted by $$Lt_{x=a} f(x)$$, or by $$\lim_{x=a} f(x)$$.

If $$f(x)$$ is a function of one variable $$x$$ in a domain which extends to infinite values, and if, after $$\epsilon$$ has been specified, it is possible to find a number $$N$$, so that $$|f(x') - f(x)| < \epsilon $$ for all values of $$x$$ and $$x'$$ which are in the domain and exceed $$N$$, then there is a number $$L$$ which has the property that $$|f(x) - L| < \epsilon $$ for all such values of $$x$$. In this case $$f(x)$$ has a limit $$L$$ at $$x = \infty$$. In like manner $$f(x)$$ may have a limit at $$x = - \infty$$. This statement includes the case where the domain of the argument consists exclusively of positive integers. The values of the function then form a “sequence,” $$u_1, u_2, \ldots\, u_n, \ldots\ ,$$ and this sequence can have a limit at $$n = \infty$$.

The principle common to the above definitions and theorems is called, after P. du Bois Reymond, “the general principle of convergence to a limit.”

It must be understood that the phrase “$$x = \infty$$” does not mean that $$x$$ takes some particular value which is infinite. There is no such value. The phrase always refers to a limiting process in which, as the process is carried out, the variable number $$x$$ increases without limit: it may, as in the above example of a sequence, increase by taking successively the values of all the integral numbers; in other cases it may increase by taking the values that belong to any domain which “extends to infinite values.”

A very important type of limits is furnished by infinite series. When a sequence of numbers $$u_1, u_2, \ldots, u_n, \ldots$$ is given, we may form a new sequence $$s_1, s_2, \ldots, s_n, \ldots$$ from it by the rules $$s_1 = u_1, s_2 = u_1 + u_2, \ldots\ s_n = u_1 + u_2 + \ldots + u_n$$ or by the equivalent rules $$s_1 = u$$ [sic], $$s_n - s_{n-1} = u_n (n=2, 3, \ldots)$$. If the new sequence has a limit at $$n = \infty$$, this limit is called the “sum of the infinite series” $$u_1 + u_2 + \ldots$$, and the series is said to be “convergent” (see ).

A function which has not a limit at a point $$a$$ may be such that if a certain aggregate of points is chosen out of the domain of the argument, and the points $$x$$ in the neighbourhood of $$a$$ are restricted to belong to this aggregate. [sic] then the function has a limit at $$a$$. For example, $$\sin(1/x)$$ has limit zero at $$0$$ if $$x$$ is restricted to the aggregate $$1 / \pi, 1 / 2\pi, \ldots, 1 / n\pi, \ldots$$ or to the aggregate $$1 / 2\pi, 2 / 5\pi, \ldots, n / (n^2 +1)\pi, \ldots$$ but if $$x$$ takes all values in the neighbourhood of $$0$$, $$\sin(1/x)$$ has not a limit at $$0$$. Again, there may be a limit at $$a$$ if the points $$x$$ in the neighbourhood of $$x$$ are restricted by the condition that $$x - a$$ is positive; then we have a “limit on the right” at $$a$$; similarly we may have a “limit on the left” at a point. Any such limit is described as a “limit for a restricted domain.” The limits on the left and on the right are denoted by $$f(a-0)$$ and $$f(a+0)$$.

The limit $$L$$ of $$f(x)$$ at $$a$$ stands in no necessary relation to the value of $$f(x)$$ at $$a$$. If the point $$a$$ is in the domain of the argument, the value of $$f(x)$$ at $$a$$ is assigned by the rule of calculation, and may be different from $$L$$. In case $$f(a) = L$$ the limit is said to be “attained.” If the point $$a$$ is not in the domain of the argument, there is no value for $$f(x)$$ at $$a$$. In the case where $$f(x)$$ is defined for all points in an interval containing $$a$$, except the point $$a$$, and has a limit $$L$$ at $$a$$, we may arbitrarily annex the point $$a$$ to the domain of the argument and assign to $$f(a)$$ the value $$L$$; the function may then be said to be “extrinsically defined.” The so-called “indeterminate forms” (see ) are examples.

7. Superior and Inferior Limits; Infinities.—The value of a function at every point in the domain of its argument is finite, since, by definition, the value can be assigned, but this does not necessarily imply that there is a number $$N$$ which exceeds all the values (or is less than all the values). It may happen that, however great a number $$N$$ we take, there are among the values of the function numbers which exceed $$N$$ (or are less than $$-N$$).

If a number can be found which is greater than every value of the function, then either there is one value of the function which exceeds all the others, or  there is a number $$S$$ which exceeds every value of the function but is such that, however small a positive number $$\epsilon$$we take, there are values of the function which exceed $$S - \epsilon$$. In the case the function has a greatest value; in case  the function has a “superior limit” $$S$$, and then there must be a point $$a$$ which has the property that there are points of the domain of the argument, in the neighbourhood of $$a$$ for any $$h$$, at which the values of the function differ from $$S$$ by less than $$\epsilon$$. Thus $$S$$ is the limit of the function at $$a$$, either for the domain of the argument or for some more restricted domain. If $$a$$ is in the domain of the argument, and if, after omission of $$a$$, there is a superior limit $$S$$ which is in this way the limit of the function at $$a$$, if further $$f(a) = S$$, then $$S$$ is the greatest value of the function; in this case the greatest value is a limit (at any rate for a restricted domain) which is attained; it may be called a “superior limit which is attained.” In like manner we may have a “smallest value” or an “inferior limit,” and a smallest value may be an “inferior limit which is attained.”

All that has been said here may be adapted to the description of greatest values, superior limits, &c., of a function in a restricted domain contained in the domain of the argument. In particular, the domain of the argument may contain an interval; and therein the function may have a superior limit, or an inferior limit, which is attained. Such a limit is a maximum value or a minimum value of the function.

Again, if, after any number $$N$$, however great, has been specified, it is possible to find points of the domain of the argument at which the value of the function exceeds $$N$$, the values of the function are said to have an “infinite superior limit,” and then there must be a point $$a$$ which has the property that there are points of the domain, in the neighbourhood of $$a$$ for any $$h$$, at which the value of the function exceeds $$N$$. If the point $$a$$ is in the domain of the argument the function is said to “tend to become infinite” at $$a$$; it has of course a finite value at $$a$$. If the point $$a$$ is not in the domain of the argument the function is said to “become infinite” at $$a$$; it has of course no value at $$a$$. In like manner we may have a (negatively) infinite inferior limit. Again, after any number $$N$$, however great, has been specified and a number $$h$$ found, so that all the values of the function, at points in the neighbourhood of $$a$$ for $$h$$, exceed $$N$$ in absolute value, all these values may have the same sign; the function is then said to become, or to tend to become, “determinately (positively or negatively) infinite”; otherwise it is said to become or to tend to become, “indeterminately infinite.”

All the infinities that occur in the theory of functions are of the nature of variable finite numbers, with the single exception of the infinity of an infinite aggregate. The latter is described as an “actual infinity,” the former as “improper infinities.” There is no “actual infinitely small” corresponding to the actual infinity. The only “infinitely small” is zero. All “infinite values” are of the nature of superior and inferior limits which are not attained. 8. Increasing and Decreasing Functions.—A function $$f(x)$$ of one variable $$x$$, defined in the interval between $$a$$ and $$b$$, is “increasing throughout the interval” if, whenever $$x$$ and $$x'$$ are two numbers in the interval and $$x' > x$$, then $$f(x') > f(x)$$; the function “never decreases throughout the interval” if, $$x'$$ and $$x$$ being as before, $$f(x') > f(x)$$. [sic] Similarly for decreasing functions, and for functions which never increase throughout an interval. A function which either never increases or never diminishes throughout an interval is said to be “monotonous throughout” the interval. If we take in the above definition $$b > a$$, the definition may apply to a function under the restriction that $$x'$$ is not $$b$$ and $$x$$ is not $$a$$; such a function is “monotonous within” the interval. In this case we have the theorem that the function (if it never decreases) has a limit on the left at $$b$$ and a limit on the right at $$a$$, and these are the superior and inferior limits of its values at all points within the interval (the ends excluded); the like holds mutatis mutandis if the function never increases. If the function is monotonous throughout the interval, $$f(b)$$ is the greatest (or least) value of $$f(x)$$ in the interval; and if $$f(b)$$ is the limit of $$f(x)$$ on the left at $$b$$, such a greatest (or least) value is an example of a superior (or inferior) limit which is attained. In these cases the function tends continually to its limit.

These theorems and definitions can be extended, with obvious modifications, to the cases of a domain which is not an interval, or extends to infinite values. By means of them we arrive at sufficient, but not necessary, criteria for the existence of a limit; and these are frequently easier to apply than the general principle of convergence to a limit (§ 6), of which principle they are particular cases. For example, the function represented by x log (1/x) continually