The necessary and sufficient condition for the existence of limit is that both left and right limits exist and are equal.
Expressed by mathematical expression as:
Conditions for the nonexistence of limit:
1, when one or both of the left limit and the right limit do not exist;
2. Both left and right limits exist, but they are not equal.
Extended data
To find the limit of a specific sequence, you can refer to the following methods:
1, using monotone bounded convergence criterion to find the limit of sequence.
Firstly, the monotonicity and boundedness of sequence are judged by mathematical induction or inequality scale method, and then the existence of limit is judged. Secondly, by taking the limit in the recursive relation and solving the equation, the limit value of the sequence is obtained.
2. Use function limit to find the limit of sequence.
If sequence limit can be regarded as a special case of function limit, then the relationship between function limit and sequence limit can be transformed into finding function limit, and then it can be solved by Robida's law.
3. There are several ways to find the limit of the sum or product series of n terms:
(1) Use special series summation method.
If the general term in the term and limit can be eliminated by dislocation or can be transformed into some form of known limit, then the limit result can be obtained directly by sorting.
(2) Using power series summation method
If we can find the power series corresponding to this series, we can find its corresponding sum function by the method of power series function, and then substitute this limit form into the corresponding variable to find the function value.
(3) Use the definition of definite integral to find the limit.
If each term of a series can be given a factor and the rest can be expressed by a general term, then the definition of definite integral can be considered to solve the limit of the series.
(4) Using the pinch theorem to find the limit.
If each term of a series can be given a factor, and the remaining items cannot be expressed by a general term, but the remaining items are arranged in ascending or descending order, we can consider using the pinch theorem to solve it.
(5) Find the limit of the product of n series.
Generally, the sum of terms is in the form of logarithm, and then it is calculated by solving the limit of terms and series.
Baidu Encyclopedia-Function Limitation