Substitution method: if the function is continuously derivable at this point, when it reaches this point, it can be substituted into the original function expression for calculation.

Limit is the basic concept of calculus and a branch of mathematics. The "limit" in a broad sense is "infinitely close and never reached". The "limit" in mathematics refers to a variable in a function, which is always in the process of getting bigger (or smaller) and changing.

In the process of approaching a certain value a gradually, "it can never coincide with a" ("it can never be equal to a, but it is enough to obtain a high-precision calculation result).

The change of this variable is artificially defined as "always approaching", and it has a "tendency to approach point A". Limit is a description of the state of change. The value a that this variable always approaches is called the "limit value" (of course, it can also be expressed by other symbols).

The above is a popular description of the connotation of "limit", and the strict concept of "limit" is finally strictly expounded by Cauchy and Weisstras.

Introduction:

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 the problem with limit thought can be summarized as follows: for the unknown quantity to be investigated, first try to correctly conceive another variable related to its change, and confirm that the trend result of this variable is very accurate through the infinite change process, which is approximately equal to the unknown quantity; 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 (finding the maximum or minimum value of 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.