What are the various formulas for finding the limit? What is the limit formula?

An equivalent infinitesimal.

The x in the three positions of the equation is replaced by the same function.

e^x-1～x

x→0)，e^(x^2)-1～x^2

x→0)。1-cosx～1/2x^2

x→0)，1-cos(x^2)～1/2x^4x→0)。1、e^x-1～xx→0)

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 abxx 0)14, [(1+x) 1 n]-1 1 nxx 0)15, loga(1+x) x lna(x 0) extended data; 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. The equivalent infinitesimal substitution is used to find the limit, and the original formula can be simplified and calculated.

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.

7. Use two important limit formulas to find the limit.

8. Use the left and right limits to find the limit, (often for finding the limit value at a break point) 9. Find the limit by Lopida's rule.

2. Find the limit formula (2) (3) (4) (5) (6) (7) (8) 3. Method (1) When the denominator limit is 0, decompose the factor, and make up equation (2) at that time, divide by xn (3) of the highest exponent and replace sinx x by the equivalent infinitesimal quantity; tan～x; arctanx～x; arcsinx～x;Derivative: (1)(c).'=0（2）（xμ）'=μxμ-1 （3）（4） （5）（ax）'=axlna（a＞0,a≠1）（6）（ex）'=ex （7）（8） （9）（sinx）'=cosx（10）（cosx）'=-sinx （11）（12） （13）（secx）'=secx·tanx（14）（cscx）'=-cscx·cotx （15）（16） （17）（18） 2.The Four Laws of Derivatives Let u=u(x) and v=v(x) be derivatives of x, then there is (1)(u v).'=u'±v' （2）（u·v）'=u'·v+u·v' （3）（cu）'=c·u' （4） （5） （6）（u·v·w）'=u'·v·w+u·v'·w+u·v·w'

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 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.

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 and the limit of 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.

Other formulas: 1. The exact calculation of the elliptic circumference (L) requires the summation of integrals or infinite series, which was first proposed by Bernoulli and developed by Euler, and the discussion of such problems led to the (0 - pi 2) integral of elliptic integral l = 4a * sqrt(1-e sin t), where a is the major axis of the ellipse and e is the eccentricity.

2. Approximate calculation of definite integrals, application of related formulas for definite integrals, spatial analytic geometry and vector algebra, differential method of multivariate functions and its application, application of differential method in geometry, direction derivative and gradient, extreme values of multivariate functions and their calculations, reintegration and its application, cylindrical coordinates and spherical coordinates, curve integral, surface integral, Gaussian formula, Stokes formula is the relationship between curve integral and surface integral.

3. Let it be a set of infinite real number series 2113. If there is a definite 5261 real number a, n>0 for any positive 4102 number, and if the limit of the series exists, then the limit value is unique, and the limit of any of its subcolumns is equal to that of the original series. There is jujube brightness:

If there is a limit to the convergence of a sequence), then the sequence must be bounded.

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.

The 9 formulas commonly used for the limit are: e x-1 x (x 0), e (x 2)-1 x 2 (x 0), 1-cosx 1 2x 2 (x 0), 1-cos(x 2) 1 2x 4 (x 0), sinx x(x 0), tanx x (x 0), arcsinx x(x 0), arctanx x(x 0), 1-cosx 1 2x 2 (x 0).

"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" in mathematics refers to the process in which a variable in a function gradually approaches a certain definite value a in the process of becoming larger (or smaller) forever and "can never coincide to a" ("eternal family bend cannot be equal to a, but taking equal to a' is enough to obtain high-precision calculation results").

The change in this variable is artificially defined as "always approaching without stopping", and it has a "tendency to constantly move towards 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 main thinking steps of limit calculation:

When we get a limit, the most important thing is to determine the type of limit, that is, which of the 7 infinitive forms it belongs to. Each infinitive has its own unique way of solving the problem, so it is especially important to determine the type. The method of judging is also very simple, and you can judge the specific type by directly bringing in the approximation value.

Factors that are non-zero constants are brought in directly by the limit signs, because they do not belong to the category of infinitesimal or infinitesimus. Equivalent infinitesimal substitution, replace everything that can be replaced first. As a reminder, the substitution must be the whole equation together and can be substituted. <>