How did the two important limit formulas in advanced mathematics come about?

Both can be proved by the definition of derivatives, or by Lopida's law.

The first one is the derivative of sinx at (0,0). The second takes the logarithm first.

in, is the derivative of in(x+1), which is calculated to be 1 and the result is e 1.

1. Use definitions to find the limit

For example: a lot of it doesn't have to be written!

2. Use the Cauchy criterion to seek!

Cauchy's criterion: To make the limiting sufficient condition such that any is given to >0, there is a natural number n, such that when n > n, for.

Any natural number m has |xn-xm|<ε

3. Use the arithmetic properties of the limit and the known limit to find it!

For example: lim(x+x

lim(x^

4. Use the inequality that is: the clamping theorem!

I won't give you any examples!

5. Use variable substitution to find the limit!

For example, limx 1 m-1) (x 1 n-1) can make x=y mn

Got: n m

6. Use two important limits to find the limit.

1）limsinx/x=1

x->0

2)lim1+1/n)^n=e

n->∞

7. There must be a limit to the use of monotonous boundaries!

8. Use the continuous property of the function to find the limit.

9. Use the law of Lobida, which is the most used.

10. Use Taylor's formula to find it, which is often used.

There are no eight important limit formulas for high numbers, only two for land.

1. The first important limit of the Gong Nian Hunger Formula:

lim sinx x = 1 (x->0) when x 0, the limit of sin x is equal to 1; Note that 1 x is infinitesimal at x.

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.

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

2. Boundedness: If a series of numbers converges (there is a limit), then the series must be bounded. However, if a series of numbers is bounded, the series may not converge.

3. Relationship with subcolumns: the sequence is the same as any of its ordinary subcolumns that converge or diverge, and have the same limit at convergence; Sequences of sufficient and necessary conditions for convergence.

Yes: Any non-trivial subcolumn of the sequence converges.

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 real number a of 5261, n>0 for any positive 4102 number, the uniqueness If the limit of the series exists, then the limit value is unique, and the limit of any of its subcolumns is equal to the original number of the series. Boundedness:

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

The formula for the second important limit: lim (1+1 x) x = e(x) When x, the limit of (1+1 x) x is equal to e;

Or when x 0, the limit of (1+x) (1 x) is equal to e.

The second depends on the occasion, in the overall multiplication and division operation, the equivalent infinity can be substituted, but the addition and subtraction operation cannot be replaced. It cannot be substituted in finding the limit of the power finger function, because when taking the logarithm, division becomes subtraction, and multiplication becomes addition.

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 obtained according to the property of infinitesimal is 0.

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

What do these two important limits do? The usefulness of these two important limits is simply too great:

1) The limit of sinx x is often distorted as equivalent infinitesimal in the teaching environment in China.

And in the international calculus.

In teaching, it is still decent, and there is no such thing as a domestic crazy speculation equivalent infinitesimal substitution. After sinx goes through the McLaughlin series, x is the infinitesimal of the lowest price, sinx is only in ratio to x, and when x tends to 0, the limit is 1. In our usual and not very appropriate expression, it is "to replace the song with the straight".

This property is used in the calculation and derivation of other limit formulas, derivative formulas, and integral formulas.

, will be used over and over again. sinx, x, and tanx also provide the most primitive examples of the clamp and squeeze theorem, as well as complex variable functions.

The definite integral of sinx x in .

Provide image understanding.

2) The importance of e is even more extreme. On the surface it serves two purposes:

a. There is a common limit between an ascending and descending number of classes and a descending number;

b. Shattered some of our original preconceived notions:

The result of an infinite power of a number greater than 1 will get smaller and smaller until 1; A positive number less than 1 will have an infinite power result that grows larger and larger until 1.

On the whole, the important limits of e have several significance:

a. Algebraic functions and logarithmic functions.

Trigonometric functions, integrated into a holistic theory, combined with complex number theory, become a complete theoretical system that is closely interconnected, complementary, complementary, and mutually corroborating.

b. Make the whole theory of calculus, including the theory of differential equations, concise and clear. Without the function e x, there would be no lnx, there would be no theory, and all formulas would be very complex.

The two important limit formulas for Advanced Mathematics are as follows:

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 obtained according to the property of infinitesimal 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 "limit" in mathematics refers to a variable in a function, which gradually approaches a certain definite value a in the process of becoming larger (or smaller) forever, and "can never coincide to a".

How to find the limit:

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.

1. For continuous elementary functions, the limit can be found within the scope of the defined domain, and the point can be directly annihilated 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. The equivalent infinitesimal substitution is used to find the limit, and the original formula can be simplified and calculated.

6. Using the two limits of the existence criterion to find the limit, some problems of Natangan can also be considered to use zoom and zoom, and then use the method of the clamping theorem to find the limit.

There are no eight important limit formulas for high numbers, only two.

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.

Possessive properties: 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. Pei Gaoqing.

2. Boundedness: If a series of numbers converges (there is a limit), then the series must be bounded. However, if a series of numbers is bounded, the series may not converge.

3. The relationship with the sub-column: the number series is the same as any of its ordinary sub-columns that converge or diverge, and has the same limit when converging; The sufficient and necessary condition for the convergence of the sequence is that any nontrivial sub-column of the sequence converges.

Two important limits:

<> 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 said to be the limit of the sequence, or the sequence converges at a.

If the above conditions do not hold the Zheng family, that is, there is a certain positive number, no matter how many positive integers n is, there is a certain n>n, such that |xn-a|a, it is said that the series does not converge to a. If it does not converge to any constant, it is called divergence.

There are no eight important limit formulas for higher numbers, only two of them.

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;

In particular, 1 x is infinitesimal and the limit of infinitesimal properties 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 first formula for the limit of the macro beat is: lim((sinx) x)=1(x->0).

The second essential important limit is lim(1+(1 x)) x=e(x).