First, let's understand the concept of limit. For a function f(x), when the independent variable x infinitely approaches a certain value A, the value of the function f(x) will infinitely approach a certain value L. We call this value the limit of the function f(x) at point A, and write it as LIMF (x) = L. The "infinite approximation" here can be understood as the difference between x and a approaches 0, that is, |x-a|.
The concept of limit has the following important properties:
1. Uniqueness: If the limit of a function exists at a certain point, then this limit is unique.
2. Locality: the limit of a function at a certain point only depends on the function value near the point, and has nothing to do with other function values far away from the point.
3. Four arithmetic rules: the limits of sum, difference, product and quotient (denominator is not 0) of two functions are respectively equal to the sum, difference, product and quotient of the limits of these two functions.
4. Pinch theorem: If the limit of a function in an interval is clamped by two functions, then the limit of this function in this interval is equal to the limit of these two functions.
Limit is widely used in mathematics. Here are some examples:
1. calculus: both derivatives and integrals are defined by limits. The derivative represents the tangent slope of the function at a certain point, and the integral represents the area of the function in an interval.
2. Series: Series is the process of adding a series of numbers. By studying the limit of series, we can judge whether the series converges and how to calculate its sum.
3. Function approximation: By studying the limit of a function at a certain point, we can find a simple function (such as polynomial, trigonometric function, etc.). ) to approximate complex functions, thus simplifying calculation and analysis.
4. Probability theory: Random variables and their distribution functions in probability theory can be described by limits. For example, the expected value and variance of discrete random variables are calculated by taking the limit of random variables.