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>… - Augustus De Morgan
" "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.
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About Augustus De Morgan
Augustus De Morgan (June 27 1806 – March 18 1871) was an Indian-born British mathematician and logician; he was the first professor of mathematics at University College London. He formulated De Morgan's laws and was the first to introduce the term, and make rigorous the idea of mathematical induction. De Morgan crater on the Moon is named after him.
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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.
Modern discoveries have not been made by large collections of facts, with subsequent discussion, separation, and resulting deduction of a truth thus rendered perceptible. A few facts have suggested an hypothesis, which means a supposition, proper to explain them. The necessary results of this supposition are worked out, and then, and not till then, other facts are examined to see if their ulterior results are found in nature.
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Spinoza's Philosophia Scripturæ Interpres, Exercitatio Paradoxa, printed anonymously ...is properly paradox, though also heterodox. It supposes, contrary to all opinion, orthodox and heterodox, that philosophy can... explain the Athanasian doctrine so as to be at least compatible with orthodoxy. The author would stand almost alone, if not quite; and this is what he meant.
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