When... we have a series of values of a quantity which continually diminish, and in such a way, that name any quantity we may, however small, all the values, after a certain value, are severally less than that quantity, then the symbol by which the values are denoted is said to diminish without limit. And if the series of values increase in succession, so that name any quantity we may, however great, all after a certain point will be greater, then the series is said to increase without limit. It is also frequently said, when a quantity diminishes without limit, that it has nothing, zero or 0, for its limit: and that when it increases without limit it has infinity or ∞ or 1⁄0 for its limit.
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Take a unit, halve it, halve the result, and so on continually. This gives—1 1⁄2 1⁄4 1⁄8 1⁄16 1⁄32 1⁄64 1⁄128 &c.Add these together, beginning from the first, namely, add the first two, the first three, the first four, &c... We see then a continual approach to 2, which is not reached, nor ever will be, for the deficit from 2 is always equal to the last term added.
...We say that—1, 1 + 1⁄2, 1 + 1⁄2 + 1⁄4, 1 + 1⁄2 + 1⁄4 + 1⁄8, &c. &c.is a series of quantities which continually approximate to the limit 2. Now the truth is, these several quantities are fixed, and do not approximate to 2. ...it is we ourselves who approximate to 2, by passing from one to another. Similarly when we say, "let x be a quantity which continually approximates to the limit 2," we mean, let us assign different values to x, each nearer to 2 than the preceding, and following such a law that we shall, by continuing our steps sufficiently far, actually find a value for x which shall be as near to 2 as we please.
[E]ach several want is limited, and... with every increase in the amount of a thing which a man has, the eagerness of his desire to obtain more of it diminishes; until it yields place to the desire for some other thing, of which perhaps he hardly thought, so long as his more urgent wants were still unsatisfied. There is an endless supply of wants, but there is a limit to each separate want. This familiar and fundamental law of human nature may pass by the name of the Law of Satiable Wants or the Law of Diminishing Utility.
It may be written thus:—
The Total Utility of a commodity to a person (...the total benefit or satisfaction yielded ...by it) increase with every increase in his stock of it, but not as fast as his stock increases. ...[i.e.,] the additional benefit ...from an additional increment of his stock of anything, diminishes with every increase in the stock ...
Something that's everything lacks limitation. There are advantages to not being able to do things. If you had everything you wanted at every moment at your fingertips, then there's nothing. There's no story. It's like Superman being able to bounces hydrogen bombs off of him. The whole series died because he didn't have any flaws. There's no story without limitation.
The following is exactly what we mean by a <small>LIMIT</small>. ...let the several values of x... bea<sub>1</sub> a<sub>2</sub> a<sub>3</sub> a<sub>4</sub>. . . . &c.then if by passing from a<sub>1</sub> to a<sub>2</sub>, from a<sub>2</sub> to a<sub>3</sub>, &c., we continually approach to a certain quantity l [lower case L, for "limit"], so that each of the set differs from l by less than its predecessors; and if, in addition to this, the approach to l is of such a kind, that name any quantity we may, however small, namely z, we shall at last come to a series beginning, say with a<sub>n</sub>, and continuing ad infinitum,a<sub>n</sub> a<sub>n+1</sub> a<sub>n+2</sub>. . . . &c.all the terms of which severally differ from l by less than z: then l is called the limit of x with respect to the supposition in question.
After these came to be relished, an infinite scale of infinites and s (ascending and descending always by infinite steps) was imagined and proposed to be received into geometry, as of the greatest use for penetrating into its abstruse parts. Some have argued for quantities more than infinite; and others for a kind of quantities that are said to be neither finite nor infinite, but of an intermediate and indeterminate nature.
Plato... introduces two infinities, because both in increase and diminution there appears to be transcendency, and a progression to infinity. Though... he did not use them: for neither is there infinity in numbers by diminution or division; since unity is a minimum: nor by increase; for he extends number as far as to the decad.
[T]he infinite is in capacity. That, however, which is infinite in capacity is not to be assumed as that which is infinite in energy. ...[I]t has its being in capacity, and in division and diminution. ...[I]t is always possible to assume something beyond it. It does not, however, on this account surpass every definite magnitude; as in division it surpasses every definite magnitude, and will be less.
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