For a given sequence a₁, a₂, a₃, ..., a_n, ..., if the value of a_n is arbitrarily close to A as n approaches a, A is the limit of the given sequence. Its notation is

     

This definition can be stated as:
A is the limit of sequence a_n, if and only if, for any chosen small positive number ε, there exists a positive number δ such that, whenever

     

then

     

The concept of limit provide a way to solve such kind problem:

The term of a sequence is getting closer and closer to a fixed number, but it can not get to that number no matter how close they are.

This concept is useful in real situation, for example: using the "method of exhaustion" to find the area of a circle.