Let {an} be a series and a be a constant. For any positive integer ε, there is always a positive integer n, which makes n >: when n (or n≥N) has | an-a | < ε (or |an-a|≤ε), the sequence {an} converges to a, and the fixed number A is called the limit of the sequence {an}, which is recorded as lim (n->. ∞) an = a. Correspondingly, the definition of sequence divergence is given.

The function limit has a definition that tends to infinity: let f be a function defined on [a, +∞) and a be a definite number. If given ε >; 0, with a positive number M(≥a), so that when x >; M, with | f (x)-a | < ε, the function f is called. When x tends to +∞, take a as the limit and write it as lim (x->; +∞) f (x) = A. There are corresponding definitions of tending to negative infinity and tending to infinity.

In addition, there is a definition that the function limit tends to x0: let f be in a hollow neighborhood u (x0; δ'), a is a fixed number. If given ε >; 0, with a positive number δ (

Nature of the restriction:

Local boundedness: if lim (x->; X0)f(x) exists, then f is bounded in the hollow neighborhood u (X0) of x0. Local number saving: if lim (x->; x0)f(x)= A & gt； 0 (or

Inequality preservation: if lim (x->; X0)f(x) and lim (x->; X0)g(x) all exist, and in a certain neighborhood u (x0; δ) If f(x)≤g(x), then lim (x->; x0)f(x)≤lim(x-& gt； x0)g(x).

Compulsive attribute: let lim (x->; x0)f(x)= lim(x-& gt； X0)g(x)=A, and in a u (x0; δ′) contains: f(x)≤h(x)≤g(x), then lim (x->; x0)h(x)=A .