The second important limit formula is lim (1+(1/x)) x = e (x →∞).

General steps of solving problems with limit thought

For the unknown quantity under investigation, first try to correctly conceive another variable related to its change, and confirm that the trend result of the' influence' of this variable through the infinite change process is very accurate and approximately equal to the unknown quantity sought; The result of the unknown quantity under investigation can be calculated by the limit principle. ?

The idea of limit is the basic idea of calculus and a series of important concepts in mathematical analysis, such as continuity of function, derivative (maximum value is 0), definite integral and so on. These are all defined by the way of limit. If you want to ask, "What is the theme of mathematical analysis?" Then it can be simply said: "Mathematical analysis is a subject that studies functions with limit thought, and the error of calculation results is too small to imagine and can be ignored.

Extreme thinking method is an indispensable and important method in mathematical analysis and even all higher mathematics, and it is also the development of "mathematical analysis" and "elementary mathematics" with continuity and further thinking. Mathematical analysis can solve many problems that elementary mathematics can't solve (such as finding instantaneous velocity, curve arc length, curved edge area, surface volume, etc.). ), it is precisely because it adopts the thinking method of "limit" and "infinite approximation" that it can get extremely accurate calculation answers. ?

People can scientifically determine the extremely accurate value of a certain quantity by investigating a series of countless more and more accurate approximation trends of some functions, which requires the use of the concept of limit and the thinking method of the above limit. It is considered that it is scientific to use the thinking method of limit, because extremely accurate conclusions can be obtained through the function calculation method of limit.

The Function of Two Important Limit Formulas

The limit of sinx/x is often misinterpreted as equivalent infinitesimal in domestic teaching environment. In the teaching of international disciplines, it is still quite satisfactory, and there is no equivalent infinitesimal substitution of crazy speculation in China. After the expansion of sinx by McLaughlin series, X is the infinitesimal of the lowest price. Only when sinx and X are in the ratio and X tends to 0, the limit is 1. In our usual, inappropriate way, it is "to replace the song directly." ? This characteristic will be used repeatedly when calculating and deducing other limit formulas, derivative formulas and integral formulas. Sinx, x and tanx also provide the most primitive examples for the compression theorem, and also provide a vivid understanding for the definite integral of sinx/x in complex variable functions. ?

The importance of e is even more extreme. On the surface, it plays two roles:

A, the rising class number has the same limit as the falling class number; ?

B, shattered some of our original inherent concepts:? The result of opening an infinite power of a number greater than 1 will be smaller and smaller until1; The positive number less than 1 will be raised to infinite power, and the result will be bigger and bigger until 1.

Generally speaking, the important limit of e has several meanings:

A. Algebraic function, logarithmic function and trigonometric function are integrated into a whole theory, and then combined with complex number theory to become a rigorous, complementary, complementary and mutually verified complete theoretical system.

B, make the whole calculus theory, including differential equation theory, concise and clear. Without the function of e x, there would be no lnx, no theory, and all formulas would be very complicated.