Limit Four Algorithm Problem, What is the Limit Four Rule?

The limit four operations can be generalized to any finite limits, but not to an infinite number of limits

Isn't it written in the book that there are two limits of addition, subtraction, multiplication and division?

When a finite limit is added, subtracted, multiplied, and divided, the result can be modeled after two, but an infinite number is not.

For example, 1 n+2 n+3 n+......n n=(n+1) 2, which is infinite when n tends to infinity. Because it is the sum of infinite terms.

But if you apply the sum of the limits, it is immediately unclear. There are infinite numbers that are 0. That's it.

For example. f(n)=n/n

lim(n->＋f(n)＝1

f(n)=n/n

1/n+1/n+……1 n (n 1 n).

lim(n->＋1/n+1/n+……1/n]=0+0+……0=1???

Contradictions, that's what it's all about.

Finite infinitesimal is infinitesimal, but infinitesimal is not necessarily infinitesimal, as in n 1 n, n infinitesimal.

Don't understand send me a message.

When a finite number of approximations are good and 0 values are added to = 0

When infinite numbers close to good and 0 are added = infinity.

1 infinity = 0

Infinity Infinity is not necessarily 0

This means that the sum of infinite numbers with a limit of 0 is exactly what is?

lim(a+b)lima+limblim(a-b)=lima-limb

limab=lima×limb

lim(a/b)lima/limb

There are many ways to find the limit

1. Continuous elementary functions.

In defining the domain. to find the limit in the range, you can substitute the point directly into the limit value, because it is a continuous function.

The limit value is equal to the value of the function at that point.

2. Use the identity deformation to eliminate the zero factor (for the 0 0 type).

3. Use the relationship between infinity and infinitesimal to find the limit.

4. Use the property of infinitesimal to find the limit.

5. Take advantage of the equivalent infinitesimal.

To find the limit of substitution, the original formula can be simplified and calculated by pure remainder.

6. Use the existence criterion of two limits to find the limit, and some problems can also consider using the method of enlargement and reduction, and then use the method of clamping theorem to find the limit.

ExtremeFour arithmeticThe rules of the hall are:

LimitsThe premise of the four rules of operation is twoLimitsexists when there is oneExtremely dressed up for a feastIf it doesn't exist, you can't use the four-rule algorithm. Let limf(x) and limg(x) exist, and let limf(x)=a, limg(x)=b.

The four operations refer to the four operations of addition, subtraction, multiplication and division. The four arithmetic is elementary mathematics.

It is also the basis for learning other related knowledge.

InLimitsIn the case of all cases, the limit of the and differential product quotient is equal toLimitsof the sum of the differential quotient. In mathematical terms, it is:

lim(a+b)lima+limb

lim(a-b)=lima-limb

limab=lima×limb

lim(a/b)lima/limb

The premise is each of the aboveLimitsare all there.

The premise of the four-rule algorithm is that two limits exist, and when one limit itself does not exist in limbs, the four-rule algorithm cannot be used.

Let limf(x) and limg(x) exist, and let limf(x)=a and limg(x)=b, then the following algorithm is used:

<> where, b≠0; c is a constant.

1. Uniqueness: If the limit of the series exists, the limit value is unique, and the limit of any of its subcolumns is equal to that of the original series.

2. Bounded: If a series of numbers is 'convergent' (with a limit), then the series must be bounded. However, if a series of numbers is bounded, the series may not converge. For example, the sequence: "1,-1,1,-1,......1)n+1”。

3. Number preservation: if.

or <0), then for any m (0,a) (a<0 is m (a,0)), there is n>0 so that n >n always exists.

Corresponding xn

The Four Rules of Algorithm of Limits can be simplified and calculated by using the four rules of operation when performing limit operations. Specifically, there are several rules:

1.The law of the sum of the two limits: lim (f(x) +g(x)) lim f(x) + lim g(x), i.e., the sum of the limits of the two function lag numbers is equal to the sum of the limits of each function.

2.The rule of the difference between the limits of two functions: lim (f(x) -g(x)) lim f(x) -lim g(x), i.e., the difference between the limits of two functions is equal to the difference between the limits of each function.

3.The rule of the product of two limits: lim (f(x) *g(x)) lim f(x) *lim g(x), i.e. the product of the limits of two functions is equal to the product of the limits of each function.

4.The law of quotient of two limits: lim (f(x) g(x)) lim f(x) lim g(x), where lim g(x) is not equal to 0, i.e., the quotient of the limits of two functions is equal to the quotient of the limits of each function.

These four-rule algorithms can help us simplify the problem and improve the efficiency of the calculation when calculating the limits.

In mathematics, the four rules of the limit refer to the following four basic rules that can be used when performing limit operations:

1.The Law of Sum and Difference of Limits (Law of Addition):

If lim(xa) f(x) = l and lim(xa) g(x) = m are present, then the following equation is satisfied:

lim(xa) [f(x) ±g(x)] l ± m

2.The product rule of the limit (multiplication Hu Xuzai's rule):

If lim(xa) f(x) = l and lim(xa) g(x) = m are present, then the following equation is satisfied:

lim(xa) [f(x) *g(x)] l * m

3.The quotient of the limit (the law of division):

If lim(xa) f(x) = l and lim(xa) g(x) = m are present, and m ≠ 0, the following equation is satisfied:

lim(xa) [f(x) g(x)] l m

4.The Composite Law of Limits (The Composite Law of Functions):

If there is lim(xa) f(x) =l and lim(yl) g(y) =n (or vice versa), and the function g is continuous at the point l, the following equation is satisfied:

lim(xa) g[f(x)] n

The four-rule algorithm of these limits allows us to use known limit results when calculating limits, simplifying the complex process of limiting the limits. It should be noted that the conditions for the application of these laws require that the functions involved satisfy certain continuity and definition requirements at the corresponding points or intervals. In the specific limit calculation, it is also necessary to analyze and derive according to the characteristics and operation rules of specific functions.

The proof of the four operations is not difficult, no knowledge of advanced mathematics is required, as long as the definition of the limit is combined, the following gives the proof of the four operations of the limit of the number series, the function can be pushed by yourself, I hope it can help you.

The Four Limits Rule:In the case where both limits exist, the limit of the sum product quotient is equal to the sum product quotient of the limit.

The premise of the four-rule algorithm is that two limits exist, and when one limit itself does not exist, the four-rule operation rule cannot be used. The idea of limits is an important idea of modern mathematics, mathematical analysis.

It is a discipline that studies functions based on the concept of limits and the theory of limits (including series) as the main tool.

Judgment of the existence or absence of limits:

1. If the result is infinitesimal.

Infinitesimal is substituted with 0, and 0 is also the limit of grinding.

2. If the limit of the numerator is infinitesimal, the denominator.

The limit is not infinitesimal, the answer is 0, and the overall limit before the cherry blossom tour is clear.

3. If the limit of the numerator is not infinitesimal and the limit of the denominator is infinitesimal, the answer is not positive infinity.

It is negative infinity, and the limit of the whole does not exist.

4. If the limits of the numerator and denominator are infinitesimal, the final result must be determined by the Robida method.