1, four algorithms of limit: this is the most basic algorithm of limit, which is used for addition, subtraction, multiplication and division. When the limits of two functions exist, the limits of their sum, difference, product and quotient can be solved by adding, subtracting, multiplying and dividing respectively. For example, if limf(x) exists and c is a constant, then LIMF (x) * c = climg (x), limf(x)+c=climf(x), LIMF (x)/c = climf (x)/c (x)/c.
2. Two important limit formulas: These are two very commonly used limit formulas, which can be used to simplify the problem in many cases. The first formula limsinx/x= 1(x tends to 0) is derived from the nature of trigonometric function and the definition of limit. The second formula lim (1+ 1/n) n = e (n tends to infinity) is a commonly used mathematical constant, which is called the base of natural logarithm. The proof of this formula requires the knowledge of power series expansion in higher mathematics.
The application function of seeking limit;
1. Determine the change trend of the function: By finding the limit, we can know the slope, concavity, infinity or infinitesimal behavior of the function at a certain point, and then determine the change trend of the function at that point.
2. Solving practical problems: When solving some practical problems, it is often necessary to ask the limit of a variable at a certain point or within a certain range. For example, find the changing trend of a physical quantity at a certain moment and find the asymptote of a curve.
3. Study the properties of functions: Limit is one of the important tools to study functions. By finding the limit, we can understand the continuity, differentiability and integrability of the function, so as to better understand the essence of the function.
4. Solving mathematical problems: In mathematics, limit is an important concept, which is often used to prove theorems and solve mathematical problems. For example, summation, integration and solving differential equations all need the concept and method of limit.
5. Determine the continuous point and discontinuous point of the function: By finding the limit, we can judge the continuous point and discontinuous point of the function at a certain point, so as to understand the overall properties of the function in a certain interval.