LIMIT : LIMITS, THEORY OF. The word limit implies a fixed magnitude to which another and a variable magnitude may be made as nearly equal as we please, it being impossible however that the variable magnitude can absolutely attain, or be equal to, the fixed magnitude. In this strict sense of the word there are two conditions which must be fulfilled before A can be called the limit of P : first, r must never become equal to A; secondly, r must be capable of being made as nearly equal to A as we please.
The method of limits is in reality nothing more than one way of evading the use of the word infinite in an absolute sense [INFINITE] : which may be shown as follows. If we take two common algebraical expressions, such as x and xx, or xs, there can be no objection to saying that when x= 7, 49, because 7 is a definite number, and the operation 7 x 7 is perfectly intelligible. And we may, if we please, say that when x approaches 7, approaches 49, so that if x may be made as near as we please to 7, can be made as near as you please to 49. Or, 7 being the limit of x, 49 is the limit of The preceding is superfluous, because it is more simple to say at once that e is 49 when x is 7. But suppose that x, instead of being taken at pleasure, must be determined by means of y; and let the investigation of the relation between x and y lead to then, so long as y has any finite value,x must be more than 7; nor can the assertion x= 7 be made without the implication that y is infinite. In this case then we can only say that x can be made as near as you please to 7, if we may take y as great as we please ; in which case can be made as near as you please to 49. In the language of the article infinite, we say (for abbreviation, as explained in INFINITE) that x is 7 and is 49, when y is infinite : iu the language of the present article, we say that x has the limit 7, and the limit 49, when y increases with.ont limit. We shall now translate the various illustrations given in the article just cited, from the language of infinites into that of limits.
When a is infinite, A is equal to B. If A be a fixed magnitude, read —If z increase without limit, A is the limit of B: if B be a fixed magni tude, read—If z increase without limit, B is the limit of A : if both A and B be variables, read—When a increases without limit, A and B approach to the same limit.
A finite quantity divided by an infinite quantity, is nothing. For this read—When the denominator of a fraction increases without limit, the numerator remaining the same, the fraction diminishes without limit.
Every circle is a regular polygon of an infinite number of sides. For this read—If the number of sides of a regular polygon inscribed in a circle be increased without limit, the polygon approaches without limit to the circle : or, the circle is the limit of all the regular polygons which can be inscribed in it.
When x is infinite, A and B are both infinite, but A is infinitely greater than B. For this read—When x increases without limit, A and n both increase without limit, but the ratio of A to B also increases without limit, or the ratio of B to A diminishes without limit.
When x =a, a is infinite. For this read—When x approaches with out limit to a, a increases without limit.
Two infinitely great quantities may have a finite ratio. For this read—When two quantities increase without limit, their ratio does not necessarily increase without limit, but may have a finite limit.
Two infinitely small quantities may have a finite ratio : or—when two quantities diminish without limit, their ratio does not necessarily diminish without limit, but may have a finite limit.
When A is infinitely small, B is infinitely great. For 'this read— When A diminishes without limit, 13 increases without limit.
An infinitely small arc of a curve is equal to its chord. For this read—When the arc of a curve diminishes without limit, the ratio of are the arc to the chord, or the fraction chord' approaches the limit unity.