Where ε is an arbitrarily small positive number, representing the absolute value of the difference between the function value and the limit value A; δ is the absolute value of the difference between the independent variable and the given point x0, indicating the closeness of the independent variable to the given point x0; The value of the function f(x) near the point x0. The ε-δ definition of limit is used to describe the value of function near a certain point, and it is a method to study the properties of function near a certain point.
In mathematics, the definition of limit is: for a function f(x), if there is a number A, when x approaches a certain point x0 (x0 can be a real number, a complex number or other mathematical objects), the value of f(x) approaches A, then we call A the limit of function f(x) when x approaches x0.
In order to better understand the concept of limit, we can consider some concrete examples. For example, consider the function f(x)=x2. When x approaches 0, the value of f(x) approaches 0. This is because the value of f(x) is getting closer and closer to 0 as x gets closer and closer to 0. In this example, the limit a is 0.
In addition to the limit in the real number range, there are some other concepts of limit in mathematics.
For example, for complex variable function, we can consider the situation that the complex variable point tends to infinity. In addition, there are some special limit concepts, such as convergence according to measure and weak convergence. These limit concepts are widely used in different mathematical fields.
The concept of limit is one of the foundations of mathematical analysis, which provides a powerful tool for mathematical analysis to describe the behavior and properties of functions. Through the concept of limit, we can study the continuity, differentiability and integrability of functions. In calculus, the concept of limit is the theoretical basis of differential and integral.