What does "limit" mean in mathematics?

Limit is an important concept in higher mathematics.

Limit can be divided into sequence limit and function limit, which are defined as follows.

Firstly, Liu Hui's Secant Circle is introduced. There is a circle with a radius of 1, and its area should be calculated when only the calculation method of straight side area is known. Therefore, he first inscribed a regular hexagon with an area of A 1, then inscribed a regular dodecagon with an area of A2 and an inscribed quadrilateral with an area of A3, thus doubling the number of sides. When n increases infinitely, An is infinitely close to the area of a circle, and he uses the inequality An+65438 to calculate the ninth power polygon of 3072=6*2. A & LTAN+2 [(an+1)-an] (n =1,2,3) ...) gets pi =3927/ 1250, which is about 3.14/kloc-.

Sequence restriction:

Definition: Let |Xn| be a series. If there is a constant a for any given positive number ε (no matter how small it is), there is always a positive integer n, so that when n >: N, the inequality

| Xn-a | & lt； ε

If both are true, then the constant a is the limit of the sequence |Xn|, or the sequence |Xn| converges to a and is written as lim Xn = a or Xn→a(n→∞).

Properties of sequence limit:

1. Uniqueness: If the limit of series exists, the limit value is unique;

2. Change the finite term of series without changing the limit of series.

Limits of several commonly used sequences;

The limit of a constant sequence is c.

An= 1/n is limited to 0.

An = x n The absolute value x is less than 1, and the limit is 0.

Professional definition of function limit;

Let the function f(x) be defined in the centripetal neighborhood of point X. If there is a constant A, there is always a positive number δ for any given positive number ε (no matter how small it is), so that when x satisfies inequality 0.

| f(x)-A | & lt； ε

Then the constant a is called the time limit of the function f(x) when x → x

Popular definition of function limit;

1, let the function y=f(x) be defined in (a,+∞). If the function f(x) infinitely approaches a definite constant a when x →+∞, then a is called the limit of the function f(x) when x tends to +∞. Let it be written as lim f (x) = a, x→+∞.

2. Let the function y=f(x) be defined near point A. When x approaches a infinitely (denoted as x→a), the value of the function approaches a constant infinitely, then A is called the limit of the function f(x) when x approaches a infinitely. Write lim f(x)=A, x → a.

Left and right limits of function:

1: if the function f(x) approaches the constant a infinitely when x approaches x0 from the left side of point x=x0 (that is, x < x0), it is said that a is the left limit of the function f(x) at point x0, and it is denoted as x→ x0-LIMF (x) = a. 。

2. if x is from the right side of point x=x0 (i.e. x >；; When x0) infinitely approaches the point x0, the function f(x) infinitely approaches the constant a, that is, A is the right limit of the function f(x) at the point x0, which is denoted as x→ x0+LIMF (x) = a. 。

Note: If the left and right limits of a function are different on x(0), then the function has no limit on x(0).

Properties of function limit:

Limit algorithm (or related formula):

lim(f(x)+g(x))=limf(x)+limg(x)

lim(f(x)-g(x))=limf(x)-limg(x)

lim(f(x)*g(x))=limf(x)*limg(x)

Lim (f (x)/g (x)) = LIMF (x)/LIMG (x) (LIMG (x) is not equal to 0).

lim(f(x))^n=(limf(x))^n

Only when the above limf(x) limg(x) exists can it be established.

lim( 1+ 1/x)^x =e

x→∞

Infinity and infinitesimal: