The concept of limit was originally described qualitatively in natural language based on the intuition of geometric intuition based on the viewpoint of motion. In high school we have learned the descriptive definition of the limit of a sequence :

　　Definition:  Suppose there is a sequence and a constant . If is infinitely close to when increases infinitely, then the constant is called the limit of the sequence , or the sequence converges to  , which is recorded as

 or .

　　If a sequence has no limit, it is said to be divergent：.

　　Note: The notation is often read as: when tends to infinity, tends to .

　　The above definition is called the descriptive definition of limit , which is essentially still an approximate description of the changing trend of variables in classical mathematics. At the undergraduate level, we will introduce a rigorous definition of the limits of a sequence . In fact, the difference between classical mathematics (elementary mathematics based on ancient Greek mathematics) and modern mathematics (advanced mathematics represented by calculus, etc.) lies in whether it is based on strict limit definitions .