Experience has convinced me that the proper way of teaching is to bring together that which is simple from all quarters, and, if I may use such a phr… - Augustus De Morgan

Aspiring to lead others, they have never given themselves the fair chance of being first led by other others into something better than they can start for themselves; and that they should first do this is what both those classes of others have a fair right to expect. New knowledge... must come by contemplation of old knowledge... mechanical contrivance sometimes, not very often, escapes this rule.

The following is exactly what we mean by a <small>LIMIT</small>. ...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 a_n, and continuing ad infinitum,a_n a_n+1 a_n+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.

...nor have I found occasion to depart from the plan... the rejection of the whole doctrine of series in the establishment of the fundamental parts both of the Differential and Integral Calculus. The method of Lagrange... had taken deep root in elementary works; it was the sacrifice of the clear and indubitable principle of limits to a phantom, the idea that an algebra without limits was purer than one in which that notion was introduced. But, independently of the idea of limits being absolutely necessary even to the proper conception of a convergent series, it must have been obvious enough to Lagrange himself, that all application of the science to concrete magnitude, even in his own system, required the theory of limits.