1, substitution method: the variable gradually approaches the limit value and observes the trend of the function value.
For example, find lim(2x+ 1). (x→2)
Answer: you can directly substitute x=2 and get? (2× 2+1) = 5 (2× 2+1) = 5, so lim(2x+ 1)=5.
2. Fractional decomposition method: the fractional decomposition is simplified to eliminate uncertain factors.
Example: find limx/sinx. (x→0)
Solution: decompose the fraction and get x/sinx = x/x * sinx/x =1/sinx/X. Because limsinx/x= 1, limx/sinx= 1.
3. Pinch theorem: The limit value is determined by the pinch function.
Example: Q? limxsin？ 1/x .(x→0)
Answer: since-1 is less than or equal to sin 1/x is less than or equal to 1 and -x is less than or equal to XSIN 1/X, when x tends to 0, both -x and x tend to 0, so according to the pinch theorem, Limxsin1/x =
4. Limit property: Derive and solve by using the property of known function limit.
Example: Find the power of lim (1+ 1/x) x. (x→ infinity).
A: According to the nature of the known function limit, the power of lim is (1+1/x) x = e.
Here are just some commonly used methods and examples to find the limit, and other methods may be involved in practical application, such as Robida's law and Taylor's expansion. When solving the limit, we should choose the appropriate method according to the specific situation, and pay attention to the use of mathematical properties and theorems.
The position in higher mathematics
Limit is an important concept in higher mathematics. Limit is usually used to describe the behavior of a function when it approaches a certain point. Let the function f(x) be defined in an eccentric neighborhood of point C. If there is a number L, there is a positive number δ for any given positive number ε, so that when x is between (0, δ), there is | f (x)-l | < ε, then the limit of the function f (x) at x=c is L, and it is recorded as lim (x→).
The concept of limit plays an important role in calculus and mathematical analysis. By studying the limit of a function at a certain point, we can explore the continuity, derivative and integral of the function. Limit is also used to solve mathematical problems such as infinity and infinitesimal. Through the calculation and properties of the limit, we can study the convergence of the function, the derivative and tangent of the curve and the Taylor series of the function. In mathematics, limit is a very important basic concept, which provides a tool for us to understand and study mathematical problems.