The limit of An is A, the limit of Bn is B, and the limit of Anbn is Ab

Bn has a limit, so there is n1>0, and when n>n1, bn is bounded, so |bn|0, when n>n2, |an-a|<ε/m。

The bn limit is b, so there is n3>0, and when n>n3, |bn-b|<ε/|a|。

The limits are calculus and mathematical analysis.

One of the most basic concepts of the other branches, the concepts of continuity and derivative are defined by it. It can be used to describe the tendency of the properties of elements in a sequence to change as the indicator gets larger and larger, or it can describe the independent variables of a function.

When approaching a certain value, the corresponding function value changes the trend.

The idea of limit is an important idea of modern mathematics, and mathematical analysis is a discipline that studies functions based on the concept of limit and the theory of limit (including series) as the main tool.

The so-called limit idea refers to "a mathematical idea that uses the concept of limit to analyze and solve problems".

The general steps to solve a problem with limit thinking can be summarized as:

For the unknown quantity to be examined, first try to correctly conceive another variable related to its change, and confirm that the 'influence' tendency of this variable through the infinite change process is very precise and equal to the unknown quantity sought; Using the limit principle, the results of the unknown quantities under investigation can be calculated.

Bn has a limit, so there is n1>0, and when n>n1, bn is bounded, so |bn|0, when n>n2, |an-a|<ε/m.

The bn limit is b, so there is n3>0, and when n>n3, |bn-b|<ε/|a|

Take n=max, when n>n, |anbn-ab|=|(an-a)bn+a(bn-b)|By definition, there is limanbn=ab

The limit of the set does not exist, the limit of the existence of the limit, and the limit that can be concluded must not exist, otherwise, if the limit exists, then the limit of an=(an+bn)-bn exists.

is set as a series of infinite real numbers. If there is a real number a, for any positive number, no matter how small, n>0 makes the inequality |xn-a|< is constant on n (n,+ then the constant a is the limit of the sequence, or the sequence converges at a.

Judgment of the existence of limits:

1. In fact, this method can be used in functions or sequences, the essence is the same, the following is an example of a sequence, the function is similar, if a single increase (single decrease), when there is m for any n there is xn<=m(xn>=m), the sequence xn converges.

2. Convergence is the existence of the limit, and divergence is the absence of the limit (which can be understood in this way). The application of the limit of the number series is mostly used to give the big problem of finding the limit of the recursive formula of the number series, you can pay attention to it.

It can be asserted.

Otherwise, if the limit exists, then.

an=(an+bn)-bn limit.

Suppose the an+bn limit exists, and the an=an+bn-bn exists from the lim(an+bn)-limbn limit, and the lim(an+bn)-limbn=lim(an+bn-bn)=limman of the limit is operated by the four rules of the limit, and the an limit exists.

There is no contradiction with the AN limit, so the AN+BN limit does not exist.

<>Limit" is a branch of mathematics – calculus.

The basic concept of "limit" in the broad sense means "infinitely close and never reachable". The "limit" in mathematics refers to the process of a variable in a certain function, which gradually approaches a certain definite value a in the process of becoming larger (or smaller) forever, and "can never coincide enough with the prestate to a", and the change of this variable is artificially defined as "always approaching without stopping", and it has a "tendency to constantly get extremely close to point a".

Limit is a description of a "state of change". The value a that this variable is always approaching is called the "limit value" (which can also be represented by other symbols). The above is a popular description of the connotation of "limit", and the strict concept of "limit" was eventually developed by Cauchy and Weierstras.

and others strictly elaborate.

The above information refers to the encyclopedia - Limit

See Burning Dig Liang Tupi Kuansan Zheng.

The following is an introduction to the limit: "limit" is the basic concept of liquid excitation calculus, a branch of mathematics, and "limit" in a broad sense means "infinitely close and never reachable". "Limit" in mathematics refers to:

In the process of a certain variable in a function that gradually approaches a certain definite value a and "can never coincide to a" in the process of becoming larger (or smaller) forever, the change of this variable is artificially defined as "always approaching without stopping", and it has a "tendency to constantly get extremely close to point a". Limit is a description of a "state of change". The value a that this variable is always approaching is called the "limit value" (which can also be represented by other symbols).

The above is a popular description of the connotation of "limit", and the concept of "limit" was finally rigorously elaborated by Cauchy and Weierstrass and others.

Have a great day.

If an >bn and the limit of an = a, and the limit of bn = b, then a > b, right? Here's a counterexample.

Only Bi can get a repentance then state b, do not stare at Tong must be strictly greater than.

For example: a[n]=1,b[n]=1-1 2 na[n]>b[n].

lima[n]=limb[n]=1.

an+bn has no limit to serve filial piety.

an*bn can have limits.

Example 1 (n+1) has a limit, n has no limit.

1 (n+1))*n -- old mu manuscript 1

Let an=a+bn

then (a1+a2+......Broad+an) n=a+(b1+b2+......bn)/n

When accompanies land n > n, bn is an infinitesimal quantity.

b1+b2+……bn) n is an infinitesimal quantity.

bn+1+…Lu Qiaoqing....+bn）/n

When a>b is the world, the numerator and the denominator are divided by a n

Get:[1-(b a) n] [1+(b a) n].When rolling the missing limb n tends to infinity, the limit is 1 when a

Let an=a+bn

then (a1+a2+......an)/n=a+(b1+b2+……bn) n When n > n, bn is an infinitesimal quantity.

b1+b2+……bn) n is an infinitesimal quantity.

bn+1+……bn) n so is also infinitesimal quantity.

So (b1+b2+......bn) n is an infinitesimal quantity, so the limit is also a

Solution: If there is a limit to a sequence an, then there is also a limit to its absolute value, and the magnitude is the same as the absolute value of the limit of the series an. i.e. if liman=a, then lim|an|=|a|

The proof is as follows: any > such as Huai Hui 0

Because liman=a

So there is n, and when n > n, there is always |an-a|"Minghe Again|an|=|an-a+a|≤|an-a|+|a|So there is |an|-|a|≤|an-a| .1) Again|a|=|a-an+an|≤|an-a|+|an|So there is |an|-|a|≥-an-a|..2) Derived from formulas (1) and (2).

an-a|≤|an|-|a|≤|an-a|i.e. ||an|-|a||≤an-a| <

For the above n, there is still ||an|-|a||<

According to the definition of limit, there is lim|an|=|a|

The proposition is proved to be hungry.