For a sequence, limit means that when the independent variable is infinitely close to a certain value, the function value is infinitely close to a constant. For example, the sequence {1, 2, 3, 4, ...} is infinite, because when the independent variable increases infinitely, the function value also increases infinitely.
For a function, limit means that when the independent variable is infinitely close to a certain value, the function value is infinitely close to a constant or infinity. For example, the limit of the function f (x) = x 2 is infinite, because when x increases infinitely, the function value also increases infinitely.
The definition of limit can be expressed in different ways, among which ε-δ definition and pinch theorem are the most common. The definition of ε-δ shows that if any given positive number ε has a positive number δ, so that the absolute value of the difference between the independent variable and the target value is less than ε, then the limit of the function at this point is the target value. The pinch theorem points out that if a function is sandwiched by two other functions, and the limit of these two functions at the target point is equal to the same constant, then the limit of this function at the target point is also equal to this constant.
Limit is widely used in mathematics, which can be used to solve the problems of derivative, integral and convergence of function series. At the same time, limit is also the basis of other branches of mathematics, such as differential equations and complex functions. Therefore, understanding the concept of limit is of great significance to the study and application of mathematics.