Lesson 1.6: Delta-Epsilon: The Formal Definition of a Limit

Thus far we have studied limits without concerning ourselves with the formal definition of a limit. This intuitive understanding of limits is generally sufficient for the purposes of Calculus BC, but because the study of mathematics of based on rigor, it is important to at least understand that the definition of a limit can be formalized. This proper definition, known as the delta-epsilon definition, is not on the AP exam, but you will be tested on it in class.

Let’s take the function and consider . Examining the graph of makes it intuitively clear that . A limit’s formal definition is rooted in error tolerances, so we ask ourselves, “How close does have to be to 2 so that is within 0.1 of 5?” The distance from to 2 is represented as and the distance from to 5 is , so we want to find a number (the Greek letter delta) such that if but . Equivalently, we can look for such that if .	The graph of
We can perform a final rewrite of our challenge as follows: find a such that if . Therefore, we graph , , and . We find that at and at . This gives us , or , so .	The graph of along with and
Now let’s try to get closer: find such that if . We graph , , and , finding that at and at . Therefore , or , so .	The graph of along with and

Let’s assign , the Greek letter epsilon, to the error tolerance (for example, we have been working with ). Incidentally, epsilon is used generally to denote any small number; the eccentric mathematician Paul Erdős referred to children as epsilons. If we continue this process with ever-decreasing values of , we will continue the pattern that has been established thus far: for this function . We define a limit as follows: if we state , that means that for every there exists a such that whenever .

Note that we have not yet proven that because we have not proven that , we have only observed that from a pattern. In the next lesson, we will work on proving that limit.