Approach Quotes - page 51
You are doing an excellent thing, one which will be wholesome for you, if, as you write me, you are persisting in your effort to attain sound understanding; it is foolish to pray for this when you can acquire it from yourself. We do not need to uplift our hands towards heaven, or to beg the keeper of a temple to let us approach his idol's ear, as if in this way our prayers were more likely to be heard. A god is near you, with you, and in you. This is what I mean, Lucilius: there sits a holy spirit within us, one who marks our good and bad deeds, and is our a guardian.
Seneca
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.
Augustus De Morgan
The following is exactly what we mean by a LIMIT. ...let the several values of x... bea1 a2 a3 a4.... &c.;then if by passing from a1 to a2, from a2 to a3, &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 an, and continuing ad infinitum,an an+1 an+2.... &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.
Augustus De Morgan
