In a general sense, the limit of a sequence is a value that it approaches with arbitrary closeness. Stated precisely: means that for every there exists a positive integer so that:
To take a simple example, let's find .
It's fairly obvious that as gets larger, the value of approaches 0, but how can we use the definition to help make that explicit?
Let . Then for any :
Here are some rules for the limits of sequences:
Now, look at these examples:
What is if ?
First, we can re-express the limit by dividing both numerator and denominator by :
Using the above rules, we can see that .
Since , it is clear that , which means the sequence must be decreasing as increases. Further, it is apparent that for any value of , so it follows that . This gives us such that:
The limit must then be . To see why, assume that it is not. Then we have:
Since this is clearly absurd for any , we see that the limit must be .