1. Convergence judgment of sequence: By calculating the limit of sequence, it can be judged whether the sequence converges. If the limit of the sequence exists and is finite, the sequence converges; If the limit does not exist or is infinite, the sequence diverges.
2. Derivation of general term formula of series: By observing the first few terms of series, we can guess the general term formula of series. Then, using the nature of limit and known conditions, the general term formula of sequence is deduced.
3. Sum and cumulative sum calculation of series: sum of series can be obtained by adding all items of series. Cumulative sum is the result of adding the first n items. By calculating the limit, we can get the expressions of series sum and cumulative sum.
4. Proof of limit existence of sequence: In some mathematical problems, it is necessary to prove the limit existence of a sequence. This can usually be proved by the nature of limit, squeezing theorem and monotone bounded criterion.
5. Relationship between limit and function derivative: In calculus, limit is closely related to function derivative. By calculating the limit of a function at a certain point, the derivative value of that point can be obtained.
6. Convergence and divergence judgment of series: series is the series of terms. By calculating the limit of series, we can judge whether the series converges or not. If the limit of series exists and is finite, the series converges; If the limit does not exist or is infinite, the series diverges.
In a word, limit is widely used in series, which can be used not only to judge the convergence of series, but also to deduce the general formula of series, calculate the sum and cumulative sum of series and prove the existence of limit.