The limit of a sequence is a fundamental concept in calculus. Consider the sequence a_n equals 1 plus 1 over n. As n increases, the terms get closer and closer to 1. We say the limit of this sequence is 1 as n approaches infinity.
The formal definition uses epsilon and N. For any small positive number epsilon, we can find a threshold N such that all terms beyond N are within epsilon distance from the limit. The green band shows the epsilon neighborhood around the limit, and the orange line shows the threshold N.
让我们考察数列 1/n。数列的项为1、1/2、1/3、1/4等等，趋向于零。对于任意ε大于零，我们可以选择N等于1/ε。那么对所有n大于N，都有1/n小于ε，从而证明了极限为零。
并非所有数列都有极限。考虑数列负一的n次方，它在负一和正一之间不断振荡。无论我们选择多大的N，总有一些项距离任何候选极限都很远。因此这个振荡数列不收敛，极限不存在。
总结一下，数列极限描述了数列项随着n趋向无穷时的行为。通过ε-N定义，我们可以严格证明极限的存在性。不同的数列可能收敛到同一个极限，如图中三个数列都收敛到1。数列极限是微积分和数学分析的基础，为后续学习奠定重要基础。