Some definitions of the concept of a limit

Leibniz (1684)

If any continuous transition is proposed terminating in a certain limit, then it is possible to form a general reasoning, which covers also the final limit.

Newton (1687)

The ultimate ratio of evanescent quantities ... [are] limits towards which the ratios of quantities decreasing without limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitem.

Maclaurin (1742)

The ratio of 2x + o to a continually decreases while o decreases and is always greater than the ratio of 2x to a while o is any real increment, but it is manifest that it continually approaches to the ratio of 2x to a as its limit.

DÕAlembert (1754)

The ratio [a : 2y + z] is always greater than a : 2y, but the smaller z is, the greater the ratio will be and, since one may choose z as small as one pleases, the ratio a : 2y + z can be brought as close to the ratio a : 2y as we like. Consequently, a : 2y is the limit of the ratio a : 2y + z.

Lacroix (1806)

The limit of the ratio (u₁ - u)/h ... is the value towards which this ratio tends in proportion as the quantity h diminishes, and to which it may approach as near as we choose to make it.

Cauchy (1821)

If the successive values attributed to the same variable approach indefinitely a fixed value, such that they finally differ from it by as little as one wishes, this latter is called the limit of all the others.