1, observation method: For some simple series, their limits can be determined by observation. For example, for the sequence 1, 1/2, 2/3, 3/4, ... it can be clearly seen that its limit is 1.

2. Definition method: If the number of terms n of a series tends to infinity and its general term an also tends to a constant a, it is said that the series converges to a, and a is called the limit of the series.

3. Geometric method: For some special series, you can use geometric figures to find the limit. For example, the series 1, 1/2, 1/4, 1/8, ... can be regarded as equidistant points in a square from the center to four directions. With the increase of the distance from the point, the points become denser and denser, and finally a limit point is formed.

4. Pinch method: If the previous item of a series is less than the latter item and the latter item is less than the previous item, then the series is said to be decreasing; If the previous item of a series is greater than the latter item, and the latter item is greater than the previous item, then the series is said to be increasing.

For decreasing series, if all its terms are greater than or equal to one value A and less than or equal to another value B, then its limit must be in the interval (a, b); For an increasing sequence, if all its terms are greater than or equal to one value A and less than or equal to another value B, then its limit must be in the interval [a, b].

Application of limits:

1. Financial field: In finance, limits are used to evaluate the risks and returns of a portfolio. By calculating the limit of portfolio return rate, the maximum possible loss of portfolio under different confidence levels can be determined, thus helping investors make more informed investment decisions.

2. Science and engineering: In science and engineering, limit is used to solve various practical problems. For example, in mechanical engineering, limits can be used to determine the strength and stiffness of mechanical parts to ensure their safety and reliability. In physics, limit can be used to describe the movement and change of objects, such as the distance an object moves in a short time.

3. Computer science: In computer science, limits can be used to determine the complexity and computational efficiency of algorithms. By calculating the limits of variables and parameters in the algorithm, the performance of the algorithm can be optimized and the running speed of the algorithm can be improved.

4. Mathematical analysis: In mathematical analysis, limit is used to study the properties and trends of functions. By calculating the limit of a function in its domain, the discontinuity, extremum and monotonicity of the function can be determined.

5. Statistics: In statistics, limits are used to study the distribution and trends of sample data. By calculating the limit of sample data, the confidence interval, confidence level and error range can be determined.