First, we need to understand the definition of limit. In advanced mathematics, the limit is usually represented by the symbol "lim", which describes the approximation degree of a function at a certain point or infinity. Specifically, if the value of function f(x) approaches L as X approaches A, then we say that the limit of function f(x) at point A is equal to L, and the "getting closer" here is a vague description. In fact, we need to judge whether the two numbers are close enough by certain standards. This criterion is the necessary and sufficient condition for the existence of limit: pinch theorem.
The pinch theorem means that for any given positive number ε (ε is a very small positive number, such as 0.00 1), there is a positive number δ (δ is a very small positive number, such as 0.00 1), so that when |x-a|.
Secondly, we need to understand the nature of limit. Restrictions have the following properties:
1. Uniqueness: If the limit of a function exists at a certain point, then its limit at that point is unique.
2. Local property: the limit of a function at a certain point is only related to the value of the function near that point, and has nothing to do with the value of the function at other points.
3. Four algorithms: the limit of sum, difference, product and quotient (denominator is not 0) of two functions at a certain point is equal to the limit of sum, difference, product and quotient of these two functions at that point.
4. Compound function rule: If the function g(x) is continuous at point A and the limit of function f(x) exists at point A, then the limit of compound function f(g(x)) at point A is equal to f(g(a)).
5. Infinity and infinity: when the limit of a function tends to 0 at a certain point, we call this function infinitesimal at that point; When the limit of a function tends to positive infinity or negative infinity at a certain point, we say that the function is infinite at that point.
Finally, we need to master the method of finding the limit. There are many ways to find the limit, and the common ones are as follows:
1. direct substitution method: when the limit of a function at a certain point can be directly calculated, we can directly substitute the value of that point into the function to solve it.
2. Pinch theorem method: When the limit of a function at a certain point cannot be directly calculated, we can use the pinch theorem to find a suitable ε and δ, and then solve the limit.
3. Robida's Law: When the limit form of a function at a certain point is "0/0" or "∞/∞", we can take the derivative of the function and then take the derivative until we get a solvable limit form.