What is a limit?
A limit is the value that a function approaches when gets very close to some value . We are allowed to say
if and only if we can make as close to as we desire by making sufficiently close to . The value of doesn’t matter—it could be different from or undefined for all we care. The purpose of the limit is to talk about what happens when is very close (but not equal) to .
Suppose I want to be no more than 0.00123 away from . If the limit exists, then this has to be possible by making a certain amount left or right of . The definition I gave above is informal, but you should be able to see that it is quite a bit more precise than simply saying, “it gets really close to .”
When we say claim that a limit exists, we are implying that the function approaches the limit from both sides—left and right. If it doesn’t do this, the limit does not exist. However, we can still talk about one side by itself using different notation. The left and right limits are represented by
In some cases it is easy to determine a limit just by looking at a graph. Consider the limit as approaches 5 for the following function:
Notice that . Remember, though, the actual value of the function when is equal to 5 is irrelevant to the limit as approaches 5. From the graph, you should be able to see that
and since these are not equal, does not exist.