Part 2/8:
A sequence ( a_n ) has a limit ( L ) as ( n ) approaches infinity if for every positive number ( \epsilon ) (no matter how small), there exists an integer ( N ) such that for all integers ( n ) greater than ( N ), the absolute difference ( |a_n - L| ) is less than ( \epsilon ). This means that as ( n ) increases, the terms of the sequence ( a_n ) get closer and closer to ( L ).
This precise definition removes ambiguity from more casual descriptions and establishes a clear framework for understanding the behavior of sequences at infinity.