1. On the limit of sequence
1. 1 series
In elementary mathematics, a sequence is defined as follows: a series of numbers arranged in a certain order is called a sequence. The mathematics textbook [1] defines a series: if the domain of the function f is all a set of positive integers n, it is called f: n → r or f(n), and n∈N is a series. Because the elements of a set of positive integers can be arranged from small to large, the sequence f(n).
Definition of limit of 1.2 sequence
Define 1, let {a} be a series and a be a constant. What if you give a positive number? Moss, there is always a positive integer n, so that n >: when n, there is | a-a |.
2. About the function limit
2. Functional limit at1x →∞
Definition 2 Let F be the function that defines [a, +∞] above, and A is a definite number. What if for any positive number? Moss has a positive number M(≥a), which makes x >: has | f (x)-a |.
Let f be a function defined on U(-∞) or U(∞). When x→∞ or x→∞, if the function value is infinitely close to a certain number, it is said that when x→-∞ or x →∞, a is the limit and f(x)=A or f (x) = a. 。
The function is limited to 2.2x→x X.
Definition 3 (functional limitations? Moss-δ definition) Let the function f be in the hollow neighborhood of point X, and U (x; δ'), a is a fixed number, and for any positive number ε, there is a positive number δ (
Similarly, f(x)=A and f(x)=A can be defined.
3. Similarities and differences between sequence limit and function limit and their roots.
As can be seen from the above definition, there are similarities and differences between sequence limit and function limit, and the methods of studying them are similar. When the same point is x→+∞, the types of sequence limit and function limit are completely similar and can be studied in the same way. The difference between them is that there is only one type of sequence limit, that is, the limit when n→∞; There are six types of function limit subdivision: x →+∞；; ; x→-∞； x→∞； x→x； x→x； The limit of x→x, the standard of classification is to classify according to different trends.
The similarity between the two is that they are both functions, and the sequence can be regarded as a function under special circumstances, and any different sequence takes the positive integer set as the domain; In the process of mathematical analysis, the function in the usual sense is defined in the real number range, and its domain can be a real number set or a subset of a real number set.
It is precisely because both of them are regarded as functions that the limit types are different due to different definition domains. The domain of the sequence is a set of positive integers, so the value of the independent variable is 1, 2, 3 ... and the minimum value of the independent variable is 1, so it is impossible to tend to-∞. And because all the items in the sequence must be integers, it is impossible to approximate a certain number, and the independent variable n is only. Generally, functions are discussed in the range of real numbers, so the independent variable X can approach both +∞ and -∞; If the function limit exists when the independent variable x approaches both +∞ and -∞, it is said that the function limit exists when x→∞. Similarly, because of the density of real number set, the independent variable X will approach a definite number X. According to the direction in which the independent variable X approaches X, the point X can be divided into left limit and right limit, so there are three kinds of X → X at a fixed point. x→x； X→x function limit.
To sum up, sequence is a special function. Because of the particularity of sequence as a function, the limit of sequence is relatively simple and has ideal properties, and all the properties of convergent sequences are holistic. However, all properties of the convergence function can only satisfy local properties. The real reason why they are different is that they have different scopes as functional domains. In my opinion, in order to truly understand the limits, it is necessary to study the causes of their differences in essence. The same theory can be studied by analogy, and the focus of learning should be on the differences between them, so as to understand what the differences are and why. Only by understanding the "why" can we really understand the corresponding knowledge.