What are the several important limit formulas?

The first important limit and the second important limit formula are:

The limit is calculus.

It refers to the tendency of a variable to gradually stabilize from the general to such a change in a certain process of change and the value (limit value) that it tends towards.

The concept of limits was eventually rigorously elaborated by people such as Cauchy and Weierstrass. In modern textbooks of mathematical analysis, almost all basic concepts (continuity, differentiation, integration) are based on the concept of limits.

Extended Information: There are many ways to find limits:

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 the limit value of the continuous function 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.

Substitution limit, which can simplify the calculation of the original formula.

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.

1. The formula for the first important limit:lim sinx x = 1 (x->0) When x 0, the limit of sin x is equal to 1.

Note that at x, 1 x is infinitesimal celery.

The limit given by the infinitesimal property is 0.

2. The formula for the second important limit:lim (1+1 x) x = e(x) when x, (1+1 x) the limit of x is equal to e; Or when x 0, the limit of (1+x) (1 x) is equal to e.

The basic methods for finding the limit are:1. Fractions. , the numerator and denominator are divided by the highest order to infinity.

is an infinitesimal calculation, and the infinitesimal is directly substituted with 0.

2. Infinity Root.

When subtracting the non-suspicious imitation of the poor big root formula, the molecule is rational and burning.

3. Apply the rule of Lobida.

However, the conditions for the application of Lobida's law are that infinity is greater than infinity, or infinitesimal is infinitesimal, and the numerator and denominator must also be continuously derivable.

The first important limit formula is: lim((sinx) x)=1(x->0), and the second important limit formula is: lim(1+(1 x)) x=e(x).

Steps to solve a problem with extreme thinking:

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.

The idea of limits is the basic idea of calculus, and it is a series of important concepts in mathematical analysis, such as the continuity of functions, derivatives (0 to obtain the maximum), and definite integrals, which are defined with the help of limits.

There are mainly two important limits.

For reference, please smile.

The definition of the limit is divided into four parts

>0 for any

The role of the definition is to depict that at x x0, f(x) can be infinitely close to the constant a, that is, f(x)-a can be arbitrarily small. In order to meet this requirement, it must be small enough.

There is δ>0

is the radius of this neighborhood, all the points that can be taken by x x0 are (x0-δ, x0) (x0, x0+δ) where x cannot take x0But we don't know, and we don't need to know, we don't know, and we don't need to know how big this neighborhood δ is, as long as we know that δ is a very small number.

0<∣x-x0∣<δ

For the independent variable x x0, again, x cannot take the point x0, but it can take all the points near and on both sides of x0. This involves the concept of neighborhood, which is a local concept of all points in the vicinity and on both sides centered on point x0.

∣f(x)-a∣<ε

Since can be small enough, f(x) can be infinitely close to the constant a, i.e., f(x) a, it is important to note here that although the independent variable x cannot get the point x0, the dependent variable f(x) can get a. Note that the existence of a function at the limit of a point does not matter whether the function is defined at that point or not.

The 16 important formulas of the limit function lim are as follows:1、e^x-1～x(x→0)。

2、e^(x^2)-1～x^2(x→0)。

cosx～1/2x^2(x→0)。

cos(x^2)～1/2x^4(x→0)。

5、sinx~x(x→0)。

6、tanx~x(x→0)。

7、arcsinx~x(x→0)。

8、arctanx~x(x→0)。

cosx~1/2x^2(x→0)。

10、a^x-1~xlna(x→0)。

11、e^x-1~x(x→0)。

12、ln(1+x)~x(x→0)。

13、(1+bx)^a-1~abx(x→0)。

14. Modify Wu Han [(1+x) 1 n]-1 1 nx(x 0).

15、loga(1+x)~x/lna(x→0)。

16、limα→0（1+α）1α=e。

"Limit" is the basic concept of calculus, a branch of mathematics, and "limit" in a broad sense means "infinitely close and never reachable". The limit of the kernel laugh in calculus is a basic concept, which refers to the tendency of a variable to gradually stabilize from a certain change process and the value of the tendency (limit value).

The formula for the limit is as follows:

1、lim(f(x)+g(x))=limf(x)+limg(x)；

2、lim(f(x)-g(x))=limf(x)-limg(x)；

3、lim(f(x)g(x))=limf(x)limg(x)；

4、e^x-1～x(x→0)；

cosx～1/2x^2(x→0)；

cos(x^2)～1/2x^4(x→0)；

7、loga(1+x)~x/lna(x→0)。

The lim limit operation formula is summarized, and the limit law of p> difference and product is summarized. When the limits of the numerator and the denominator both exist, and the limits of the denominator are not zero, the limit law of the quotient can be used.

How to find the limit:

1. For continuous elementary functions, the limit can be directly substituted into the limit value of the defined domain, because the limit value of the continuous function 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 extreme and carry the disadvantages of the early limit.

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

5. The equivalent infinitesimal substitution is used to find the limit, and the original formula can be simplified and calculated.

6. Use the two limits to argue the existence criterion of the finch, find the limit, and some problems can also consider Bu Hui to use zoom in and out, and then use the method of the clamping theorem to find the limit.

The two special limit formulas are as follows:

One is that when x tends to 0, sinx x=1;The other is that when x tends to 0, (1+x) 1 x)=e.

The mathematical definition of the limit is: in the process of a certain function covering a certain variable, the variable gradually approaches a certain definite value in the process of changing forever, and can never coincide into the process, the change of this variable is artificially prescribed to be forever close and not stop. Limit is a description of a state of change.

The general concept of the limit of the function: in the process of a certain change of the independent variable, if the corresponding function value is infinitely close to a certain definite number, then the definite number is called the limit of the digging function in this change process.

The function limit is one of the most basic concepts in advanced mathematics, and concepts such as derivatives are completed on the definition of function limits. Rational use of the limit properties of functions. The commonly used properties of function limits include the uniqueness, local boundedness, order-preserving and operation rules of function limits and the limits of composite functions.

Monotonic bounded criterion: A monotonic increase (decrease) of a series of numbers with upper (lower) bounds must converge. When using the above two items to find the limit of the function, it is necessary to pay special attention to the following key points.

First, we must first prove the convergence with a monotonous defined theorem, and then find the limit value. Second, the key to applying the entrapment theorem is to find the function with the same limit value, and to satisfy the limit is to tend to the same direction, so as to prove or find the limit value of the function.