Finding the limit of sequence is an important concept and basic skill in mathematical analysis, and it is also one of the key tools to solve various mathematical problems. Next, I will introduce several common methods of finding the limit of sequence in detail, and illustrate their usage and precautions through some specific examples.

1, definition method, the definition of sequence limit is a necessary and sufficient condition for sequence convergence, and it is also a basic method to judge whether sequence limit exists. The basic idea of the definition method is that by taking ε and n, for any positive integer n >；; When n, there is | an-a |

Where a is the limit of the sequence, ε is an arbitrarily small positive number, and n is a positive integer. The definition method requires that the selected n is related to ε, so that when n >; When n, the value of |an-a| is less than ε. Example 1: Find the limit solution of sequence 1/n: take ε= 1, then N= 1, when n >; At 1, there is |1/n-0 | =1/n.

2. Attribute method. The properties of sequence limit include monotone bounded theorem, pinch theorem and so on. Monotone definition theory points out that if the sequence monotonically increases (decreases), then the sequence has an upper bound (lower bound); Pinching theorem shows that if there are constants A and B, so that the first n term of a series is less than or equal to A and greater than or equal to B, then the series converges to the average of A and B. ..

Example 2: Find the limit of sequence n 2. Solution: According to the pinch theorem,12

3, indirect method, indirect method is through the use of known limit properties or conclusions, through deformation or transformation, to find the limit of the sequence. Example 3: Find the limit of sequence sin(π/n).

4. Transformation method, that is, the decomposition or deformation of the sequence is transformed into the form of known limit, so as to find the limit of the sequence.

Limit interpretation of sequence

1, the limit problem of sequence is an important content of our study, and at the same time, the limit theory is also one of the foundations of higher mathematics. As a basic concept of calculus, the limit problem of sequence is of great significance to calculus theory.

2. Monotonicity is defined. In real number system, monotone bounded sequence must have a limit. Compactness theorem, any bounded sequence must have a convergent subsequence.