In the case of a sequence of numbers, the limit is the target value that terms in the sequence get closer and closer to. This limit is not necessarily ever reached but can be approached arbitrarily close if the sequence is taken far enough.
More precisely, if the terms of an infinite series approach a definite value l in such a way that if we select some quantity, however small, we can always find a number N so that every term of the sequence after the Nth differs from l by less than the selected small quantity, then l is said to be the limit of the sequence. The limit may or may not be a member of the sequence.
The limit point of a point set (or curve) is a point such that, no matter how small a distance from it is chosen, there is a member of the set closer to it. Limit points may or may not be members of the set.