Limit (mathematics)

“ The power which is given by such an analysis, of expressing with more ease the mathematical laws of phenomena, depends in general on this, that since the calculus applies, not to the increments themselves of the proposed quantities, but to the limits of the ratios of those increments, we can always substitute for each increment any other magnitude more easy to consider, provided that their final ratio is the ratio of equality, or, in other words, that the limit of their ratio is unity. ”

“ All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives. [92] The notion of a limit, which is fundamental in the greater part of higher mathematics, used to be defined by means of quantity, as a term to which the terms of some series approximate as nearly as we please. But nowadays the limit is defined quite differently, and the series which it limits may not approximate to it at all. This improvement also is due to Cantor, and it is one which has revolutionised mathematics. ”

“ And when such processes as Newton thus deduced from the conception of a Limit, are represented by means of general algebraical symbols instead of geometrical diagrams, we have then before us the Method of Fluxions, or the Differential Calculus; a mode of treating mathematical problems justly considered as the principal weapon by which the splendid triumphs of modern mathematics have been achieved. ”

“ Thus the class of rational numbers whose squares are less than 2 has no upper limit among the rational numbers. But among the real numbers all classes have limits. Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of rationals or the continuity of the series of real numbers. ”

“ Mathematics recognizes several kinds of infinity. One of them occurs when numbers become very large. For example, the quotient of 12 divided by a number x, as x becomes closer and closer to 0, becomes indefinitely large, and the limit is called infinity and is denoted ∞. ”

“ This may suffice to show how the idea of infinity may be applied under different conditions and produce different results, without any contradiction. What is infinite under one aspect is not so under another aspect; hence we have the orders of infinities which figure so largely in mathematics; but I say again that these contradictions are not susceptible of any explanation if we attribute an absolute value to the idea of the infinite, instead of considering it as the abstract representation of the negation of limit. ”

“ The concept of equation or that of an equational 36 system, imposing restriction upon variability, is matched in life by the concept of natural and spiritual law, giving order to what were else chaotic change and providing partial freedom in lieu of none at all. What is known in mathematics under the name of limit is everywhere present in life in the guise of some ideal, some excellence high-dwelling among the rocks, an “ever flying perfect” as Emerson calls it, unto which we may approximate nearer and nearer, but which we can never quite attain, save in aspiration. ”

“ Use of the Differential Calculus as preparatory to that of the Integral. In forming the differential equations of any phenomenon whatever, it is very seldom that we limit ourselves to introduce differentially only those magnitudes whose relations are sought. To impose that condition would be to uselessly diminish the resources presented by the transcendental analysis for the expression of the mathematical laws of phenomena. ”

“ 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 diminishes when 1/e > x > 0 and x diminishes towards zero, and it never becomes negative. ”

“ The finite is that which has a limit. A limit is the term beyond which there is nothing of the object limited. The limits of a line, are the points beyond which the line does not extend; the limit of a number, is the extreme where the number stops; the limit of human knowledge, is the point to which we may arrive, but which we cannot go beyond. A limit being a negation, to deny a limit, is to deny a negation, and is consequently an affirmation.
27. It is easy to see from these examples, that a limit in the ordinary sense, expresses an idea distinct from what mathematicians define it. ”

“ Nor is it true that by the creation of an infinite multitude of points between two given points such a multitude would be an infinity within limits. For the two given points are limits, or rather terms, of a local relation, but they are no limits of the multitude, or discrete quantity, which can be placed between them; for, without altering the position of those two points, we can increase without end the number of the intervening points. As volume is not a limit of density, so the distance of two points is not the limit of the multitude that can be condensed between them. ”

“ Similarly, is read approaches the limit minus infinity, and is read , in numerical value, approaches the limit infinity.While the above notation is convenient to use in this connection, the student must not forget that infinity is not a limit in the sense in which we defined a limit on p. 11, for infinity is not a number at all. ”

“ But in the series, every term is only the half of the preceding; consequently no number of terms, however great, can be made as great as 2 by adding one more. The sum, therefore, of 1, 1⁄2, 1⁄4, 1⁄8 &c. continually approaches to 2, diminishing its distance from 2 at every step, but never reaching it. Hence, 2 is celled the limit of 1 + 1⁄2 + 1⁄4 + &c. We are not, therefore, to conclude that every series of decreasing terms has a limit. ”

“ Since this is an incommensurable number, we seek for its limits by extraction of the square root, namely 1415 the greater limit and 1414 the less limit, or more accurately the greater, and the less; for as we reduce the difference of the limits we increase the accuracy. ”

“ If mathematicians had been obliged to speak of "that to which a quantity, in increasing or diminishing, is always approaching nearer, so that the difference becomes less than any assignable quantity, but to which it never becomes exactly equal," instead of expressing all this by the simple phrase, "the limit of a quantity," we should probably have long remained without most of the important truths which have been discovered by means of the relation between quantities of various kinds and their limits. ”

“ But although we may not know how to compute the numerical value of that limit, it is of prime importance to know that a limit does exist, for otherwise the series may be non-convergent. When examining a series to determine whether or not it is convergent, the following theorems, which we state without proofs, are found to be of fundamental importance. [4]
Theorem I. If is a variable that always increases as increases, but always remains less than some definite fixed number , then as increases without limit, will approach a definite limit which is not greater than . ”

“ In case the function becomes, or tends to become, infinite at a, one or more of these limits is infinite in the sense explained in § 7; and now it is to be noted that, e.g. the superior limit of the inferior limits in diminishing neighbourhoods on the right of a may be negatively infinite; this happens if, after any number N, however great, has been specified, it is possible to find a positive number h, so that all the values of the function in the interval between a and a + h (a excluded) are less than −N; in such a case ƒ (x) tends to become negatively infinite when x decreases towards a ”

“ A given point on a line may be the limit of a set of points on the line, without its being necessary to bring in co-ordinates or measurement or anything quantitative. The cardinal number is the limit (in the order of magnitude) of the cardinal numbers 1, 2, 3, ... , ..., although the numerical difference between and a finite cardinal is constant and infinite: from a quantitative point of view, finite numbers get no nearer to as they grow larger. ”

“ What delayed the true theory of irrationals was a mistaken belief that there must be "limits" of series of ratios. The notion of "limit" is of the utmost importance, and before proceeding further it will be well to define it.
A term is said to be an "upper limit" of a class with respect to a relation if (1) has no maximum in , (2) every member of which belongs to the field of precedes , (3) every member of the field of which precedes precedes some member of . (By "precedes" we mean "has the relation to.") ”

“ All the more exact and successful parts of physical science depend upon its use, and yet the mathematicians who have created so great a body of exact truths have never decided upon the basis of the calculus. What is the nature of a limit or the nature of an infinitesimal? Start the question among a knot of mathematicians, and it will be found that hardly two agree, unless it is in regarding the question itself as a trifling one. Some hold that there are no such things as infinitesimals, and that it is all a question of limits. ”

“ Because, of two propositions, one of which must be true, it proves both impossible—it proves that space cannot have a limit, because a limit is inconceivable, and yet that it must have a limit, because unlimited space is inconceivable; it proves, therefore, that space has a limit and has no limit, which is absurd. ”

“ If the given product contains factors, then since each infinitesimal may be assumed less than the th root of , the product can be made less than itself.
(4) If is a variable which approaches a limit different from zero, then the quotient of an infinitesimal by is also an infinitesimal. For if limit , and is any number numerically less than , then, by definition of a limit, will ultimately become and remain numerically greater than . Hence the quotient , where is an infinitesimal, will ultimately become and remain numerically less than , and is therefore by (2) an infinitesimal. ”

“ The limit (in a quantum) is identical with the whole of the quantum itself. As in itself multiple, the limit is Extensive magnitude; as in itself simple determinateness (qualitative simplicity) , it is Intensive magnitude or Degree. ”

“ The actual limit of the number that can be “seen”—i.e. seen without counting or adding—depends for any individual on the shape and arrangement of the objects, but under similar conditions it is not the same for all individuals. It has been suggested that as many as six objects can be seen at once; but this is probably only the case with few people, and with them only when the objects have a certain geometrical arrangement. The limit for most adults, under favourable conditions, is about four. ”

“ Complete accuracy would only be attainable as a limit: if the appearances of Jones as we approach him tend towards a limit, that limit may be taken to be what Jones really is. It is obvious that from the point of view of physics the appearances of a thing close to "count" more than the appearances far off. We may therefore set up the following tentative definition:
The matter of a given thing is the limit of its appearances as their distance from the thing diminishes. ”

“ When an integral is to be thus defined between two limits, we call the lower of the two values the inferior limit, and the upper value the superior limit. Any integral so limited we designate as a definite integral, by way of distinguishing it from a general integral to which no limits are assigned. ”

“ When the second differential coefficient exists, or the first is differentiable, we have the relation
The limit expressed by the right-hand member of this equation may exist in cases in which ƒ′ (x) does not exist or is not differentiable. The result that, when the limit here expressed can be shown to vanish at all points of an interval, then ƒ (x) must be a linear function of x in the interval, is important. ”

“ If then we were to subtract the smaller area from the larger, we should have left as a remainder the area , which is what we want. Here we have the clue as to what to do; the definite integral between the two limits is the difference between the integral worked out for the superior limit and the integral worked out for the lower limit.
Let us then go ahead. First, find the general integral thus:
.
and, as is the equation to the curve (Fig. 52) ,
is the general integral which we must find. ”

“ Theorem III. The limit of the quotient of two variables is equal to the quotient of the limits of the separate variables, provided the limit of the denominator is not zero.
Before proving these theorems it is necessary to establish the following properties of infinitesimals.
(1) The sum of a finite number of infinitesimals is an infinitesimal. ”

“ Thus the argument fails to distinguish between a minimum and an inferior limit (see FUNCTION) . In order to prove Dirichlet's principle it becomes necessary to devise a proof that, in the case of the integral 1, there cannot be a limit of this kind. ”

“ When this Limit chances to be definable by us in terms quite other than those which the infinite series embodies, we are able to be, at once, in possession of the Limit, aware that this is the Limit of the infinite series, and able to see how the infinite series is absolutely incapable of ever attaining this its own goal. ”