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. Let's visualize this with the epsilon band around our limit.
让我们研究一个交替数列：a_n 等于负一的 n 次方除以 n。这个数列在正负值之间振荡，但越来越接近零。我们可以使用 epsilon-N 定义来证明这个极限，通过显示 a_n 的绝对值总是小于 1/n。
Now let's examine another sequence: a_n equals n over n plus 1. This sequence increases monotonically toward 1. We can rewrite it as 1 minus 1 over n plus 1, making the distance to the limit clearly visible. The proof follows the same epsilon-N pattern, showing that we can make the terms arbitrarily close to 1.
In summary, the epsilon-N definition of sequence limits provides a rigorous mathematical framework for understanding convergence. We've seen three different types of sequences: monotonic convergence, oscillating convergence, and rational function limits. Each demonstrates how sequences can approach their limits in unique ways, but all satisfy the same fundamental definition. This concept forms the foundation for more advanced topics in mathematical analysis.