This section introduces the formal definition of a limit. Many refer to this as “the epsilon-delta” definition, referring to the letters and of the Greek alphabet.
Before we give the actual definition, let’s consider a few informal ways of describing a limit. Given a function and an -value, , we say that “the limit of the function , as approaches , is a value ” if:
The problem with these definitions is that the words “tends,” “approach,” and especially “near” are not exact. In what way does the variable tend to, or approach, ? How near do and have to be to and , respectively?
The traditional notation for the -tolerance is the lowercase Greek letter delta, or , and the -tolerance is denoted by lowercase epsilon, or . One more rephrasing of “Tolerance Levels” nearly gets us to the actual definition:
The point is that and , being tolerances, can be any positive (but typically small) values satisfying this implication. Finally, we have the formal definition of the limit with the notation seen in the previous section.
Definition1.2.2.The Limit of a Function at a point.
Let be an open interval containing , and let be a function defined on , except possibly at . The statement that “the limit of , as approaches , is ” is denoted by
,
and means that given any , there exists such that for all in , where , if , then .
Note the order in which and are given. In the definition, the -tolerance is given first and then the limit will exist if we can find an -tolerance that works.
An example will help us understand this definition. Note that the explanation is long, but it will take one through all steps necessary to understand the ideas.
Before we use the formal definition, let’s try some numerical tolerances. What if the tolerance is , or in other words ? How close to does have to be so that is within units of ? That is, ? In this case, we can proceed as follows:
(Let )(Square the inequality)(Subtract from both sides)
Graph of the function . There are three points on the graph, the first is at ,, giving the point . For the second point we have , making , which gives the point . The third point found at which gives , giving the point .
There are two vertical lines near the axis, at . One which goes down and the other goes up. The length of the lines represents and are used to show that the points and are within of the point .
Graph of the equation . There are three points on the graph, the first is at ,, giving the point . For the second point we have , making , which gives the point . The third point found at which gives , giving the point .
This graph has the same vertical lines as Figure 1.2.5.(a) There are also horizontal lines near the axis, showing the distance from to is and the distance from to 4 is .
Given the tolerance , we have found an tolerance, , such that whenever is within units of , then is within units of . That’s what we were trying to find.
What if the tolerance is , i.e. ? How close to does have to be in order for to be within units of ? (In other words for ?) Again, we just square these values to get , or
What we have so far: if , then leads to being less than from and if , then being less than from . A pattern is not easy to see, so we switch to general try to determine an adequate symbolically. We start by assuming is within units of :
The “desired form” in the last step is “somethingsomething.” Since we want this last interval to describe an tolerance around 4, we have that either or , whichever is smaller:
The previous example was a little long in that we sampled a few specific cases of before handling the general case. Normally this is not done. The previous example is also a bit unsatisfying in that ; why work so hard to prove something so obvious? Many - proofs are long and difficult to do. In this section, we will focus on examples where the answer is, frankly, obvious, because the non-obvious examples are even harder. In the next section we will learn some theorems that allow us to evaluate limits analytically, that is, without using the - definition.
We are close to an answer, but the catch is that must be a constant value (so it can’t depend on ). There is a way to work around this, but we do have to make an assumption. Remember that is supposed to be a small number, which implies that will also be a small value. In particular, we can (probably) assume that . If this is true, then would imply that , giving .
We have arrived at as desired. Note again, in order to make this happen we needed to first be less than . That is a safe assumption; we want to be arbitrarily small, forcing to also be small.
We have also picked to be smaller than “necessary.” We could get by with a slightly larger , as shown in Figure 1.2.7. The outer lines show the boundaries defined by our choice of . The inner lines show the boundaries defined by setting . Note how these dotted lines are within the dashed lines. That is perfectly fine; by choosing within the dotted lines we are guaranteed that will be within of .
There are also vertical lines at ,,, and . This creates two sets of vertical lines, the outer set is at and . The inner set is at and . The out set of lines mark where , while the inner set marks .
In summary, given , set . Then implies (i.e. ) as desired. This shows that .Figure 1.2.7 gives a visualization of this; by restricting to values within of , we see that is within of .
When we have properly done this, the something on the “greater than” side of the inequality becomes our . We can refer to this as the “scratch-work” phase of our proof. Once we have , we can formally start the actual proof with and use algebraic manipulations to conclude that , usually by using the same steps of our “scratch-work” in reverse order.
We note that we could actually show that for any constant . We do this by factoring out from both sides, leaving us to show instead. By using the substitution , this reduces to showing which we just did in the last example. As an added benefit, this shows that in fact the function is continuous at all values of , an important concept we will define in Section 1.5.
This formal definition of the limit is not an easy concept grasp. Our examples are actually “easy” examples, using “simple” functions like polynomials, square roots and exponentials. It is very difficult to prove, using the techniques given above, that , as we approximated in Section 1.1.
There is hope. Section 1.3 shows how one can evaluate complicated limits using certain basic limits as building blocks. While limits are an incredibly important part of calculus (and hence much of higher mathematics), rarely are limits evaluated using the definition. Rather, the techniques of Section 1.3 are employed.
