Limit. One says that one quantity is the limit of another quantity, when the second can approach the first more closely than any given quantity, however small, without the quantity approaching, passing the quantity which it approaches; so that the difference between a quantity and its limit is absolutely inassignable.

For example, consider two polygons, one inscribed and the other circumscribed about a circle, it is evident that one can increase the sides however one wishes; and in this case, each polygon will more and more closely approach the circumference of the circle, the contour of the inscribed polygon increasing, and that of the circumscribed polygon decreasing; but the perimeter or the contour of the first will never surpass the length of the circumference, and that of the second will never be smaller than the same circumference; the circumference of the circle is thus the limit of the increase of the first polygon, and the decrease of the second.

1st. If two quantities have a limit of the same quantity, the two quantities will be equal to one another.

2nd. Let A × B be the product of two quantities A, B. We suppose that C is the limit of the quantity A, and D the limit of the quantity B; I say that C × D, the product of the limits , will necessarily be the limit of A × B, the product of the two quantities A, B.

These two propositions, that one will find demonstrated exactly in the elements of Geometry , are used as principles to show rigorously that the area of a circle, is the product of its semicircumference by its radius. See the work cited p. 331 and following in the second volume. (E)

The theory of limits is the basis of the true metaphysics of differential calculus. See Differential, Fluxion, Exhaustion, Infinity. Properly speaking, the limit never coincides, or is never equal to the quantity of which it is the limit ; but it is approached more and more, and can differ by as little as one wants. The circle, for example, is the limit of the inscribed and circumscribed polygons; because it never merges with them, though they can approach it ad infinitum. This notion can serve to clarify many mathematical propositions. For example, one says that the sum of a decreasing geometric progression whose first term is a and the second b , is
; this value is at no point strictly equal to the sum of the progression, it is the limit of this sum, that is to say the quantity which it can approach as nearly as one wants, without ever arriving at it exactly. Because if e is the last term of the progression, the exact value of the sum is
, which is always less than
, because in the same decreasing geometric progression, the last term is never = 0: but as this term continuously approaches zero, without ever arriving there, it is clear that zero is its limit , and consequently the limit of
is
, by supposing e = 0, that is to say putting in the place of e its limit . See Sequences or Series, Progression, etc. (O)