Limit is a basic concept in calculus, which refers to such a changing trend and the trend value (limit value) that variables gradually tend to be stable in a certain changing process. The concept of limit was finally strictly expounded by Cauchy and Wilstrass. In modern mathematical analysis textbooks, almost all basic concepts (continuity, differentiation and integration) are based on the concept of limit.
Limit thought is an important thought in modern mathematics, and mathematical analysis is a subject that studies functions with the concept of limit and limit theory (including series) as the main tools.
The so-called limit thought refers to "a mathematical thought that uses the concept of limit to analyze and solve problems".
The general steps to solve problems with extreme ideas can be summarized as follows:
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 thoughts are widely used in modern mathematics and even physics, which is determined by its inherent thinking function. The thought of limit reveals the unity of opposites between variables and constants, infinity and finiteness, which is the application of the law of unity of opposites of materialist dialectics in the field of mathematics. With the help of limit thought, people can know infinity from finiteness, change from invariability, curve from linear formation, qualitative change from quantitative change and accuracy from approximation.
The two concepts of "infinity" and "finiteness" are different in essence, but they are related. Infinity is the concept of abstract thinking in the brain, which exists in the brain. "finiteness" is the mapping of ever-changing things that exist objectively, and "infinity" that conforms to the laws of objective reality belongs to the whole. According to axioms, the whole is greater than the local thinking.
"Change" and "invariability" reflect two different states, namely, movement change and relative stillness, but they can be transformed into each other under certain conditions, which is "one of the powerful levers of mathematical science". For example, in physics, it is impossible to find the instantaneous speed of linear motion with variable speed by elementary methods. The difficulty lies in whether the instantaneous speed of variable-speed linear motion is variable. For this reason, people first use the "uniform" calculation method instead of the "variable speed" state calculation in a small time interval to find its average speed, and define the instantaneous speed in a small time as "speed limit", that is, by means of the limit thinking method, seek the accurate "limit" result of "change at a certain moment" from the "unchanged" form.