Introduction: Limit is the most basic method to study functions, which describes the changing trend of functions when independent variables change. From the definition of sequence limit to the definition of function limit, the key is to make it clear that sequence is also a function. The sequence can be regarded as a function whose domain is natural number set, and its analytical expression is an=f(n). Such as continuity, derivative, integral, etc. , should be defined by limit, and the limit method generated by limit is the most basic method of mathematical analysis. Better understanding of limit thought, mastering limit theory and applying limit method are the keys to continue studying mathematical analysis. This paper will mainly expound the concept, properties and methods of limit. The ε-N definition of sequence limit is the focus and core of limit theory. The definition of sequence limit is 1.

Given the sequence {an} and the constant a, if there is any given positive integer ε (no matter how small it is), there is always a positive integer n, so that when n >: when n, the inequality | an-a |.

When n tends to infinity, the limit of an is equal to a or an tends to a, and the sequence {an} is called a convergent sequence if its limit exists, otherwise it is called a divergent sequence.

The geometric meaning defined above is that for any open interval (A-ε, A+ε) with the center of a, one term an can always be found in the sequence {an}, so that all subsequent terms are located in this open interval, and outside this interval, there are only limited terms (n terms) of {an} at most.

There are two points to note about the positive integer n: first, n and ε exist, generally speaking, n increases with ε decreasing, but n is not unique; Secondly, the definition only emphasizes the existence of a positive integer n, rather than finding the smallest n. We only pay attention to the fact that all terms after the nth term can keep the distance from the constant A less than any given small positive number ε. 2. attributes.

The convergent sequence has the following properties: (1) limit uniqueness;

(2) If the sequence {an} converges, then {an} is a bounded sequence;

(3) If the sequence {an} has a limit a, any of its subsequences {ank} also has a limit a; (4) number preserving, that is, if the limit is A>0, there is a positive integer N 1, n >；; An > when n1; 0;

(5) Order preservation, that is, if, and a: An.

(1) The limit of the function when the independent variable tends to a finite value:-

[www.uuubuy.com, paper web] The function f(x) is defined in the centripetal neighborhood of point x0. If there is always a positive number δ for any given positive number ε (no matter how small), so that the corresponding function value f(x) satisfies inequalities for all X that satisfy inequalities, then the constant A is the limit of the function f(x) at x→x0, which is recorded as.

The geometric meaning of the above definition is as follows: the four segments in the limit definition are expressed in geometric language as 1 pair: an arbitrary belt-shaped region with two straight lines as the boundary; 2 Total: the radius of total existence (centered on the position of point x0);

3 At that time: when the point X is located in the δ hollow neighborhood with the bit center of the point x0; 4 is: the image of the corresponding function f(x) is located in this band.

(2) the limit of the function when the independent variable tends to infinity: let the function f(x) be defined as |x| greater than a positive number, if ε >; 0, there is always a positive number χ, so for a suitable inequality | x | >；; All x of χ and the corresponding function value f(x) satisfy the inequality | f (x)-a |.