By now you have progressed from the very informal definition of a limit in the introduction of this chapter to the intuitive understanding of a limit. At this point, you should have a very strong intuitive sense of what the limit of a function means and how you can find it. In this section, we convert this intuitive idea of a limit into a formal definition using precise mathematical language. The formal definition of a limit is quite possibly one of the most challenging definitions you will encounter early in your study of calculus; however, it is well worth any effort you make to reconcile it with your intuitive notion of a limit. Understanding this definition is the key that opens the door to a better understanding of calculus.
Before stating the formal definition of a limit, we must introduce a few preliminary ideas. Recall that the distance between two points a and b on a number line is given by | a − b | . | a − b | .
It is also important to look at the following equivalences for absolute value:
With these clarifications, we can state the formal epsilon-delta definition of the limit .
Let f ( x ) f ( x ) be defined for all x ≠ a x ≠ a over an open interval containing a. Let L be a real number. Then
lim x → a f ( x ) = L lim x → a f ( x ) = Lif, for every ε > 0 , ε > 0 , there exists a δ > 0 , δ > 0 , such that if 0 < | x − a | < δ , 0 < | x − a | < δ , then | f ( x ) − L | < ε . | f ( x ) − L | < ε .
This definition may seem rather complex from a mathematical point of view, but it becomes easier to understand if we break it down phrase by phrase. The statement itself involves something called a universal quantifier (for every ε > 0 ), ε > 0 ), an existential quantifier (there exists a δ > 0 ), δ > 0 ), and, last, a conditional statement (if 0 < | x − a | < δ , 0 < | x − a | < δ , then | f ( x ) − L | < ε ). | f ( x ) − L | < ε ). Let’s take a look at Table 2.9, which breaks down the definition and translates each part.
| Definition | Translation |
|---|---|
| 1. For every ε > 0 , ε > 0 , | 1. For every positive distance ε from L, |
| 2. there exists a δ > 0 , δ > 0 , | 2. There is a positive distance δ δ from a, |
| 3. such that | 3. such that |
| 4. if 0 < | x − a | < δ , 0 < | x − a | < δ , then | f ( x ) − L | < ε . | f ( x ) − L | < ε . | 4. if x is closer than δ δ to a and x ≠ a , x ≠ a , then f ( x ) f ( x ) is closer than ε to L. |
We can get a better handle on this definition by looking at the definition geometrically. Figure 2.39 shows possible values of δ δ for various choices of ε > 0 ε > 0 for a given function f ( x ) , f ( x ) , a number a, and a limit L at a. Notice that as we choose smaller values of ε (the distance between the function and the limit), we can always find a δ δ small enough so that if we have chosen an x value within δ δ of a, then the value of f ( x ) f ( x ) is within ε of the limit L.
left and right of point a can be found so that if we have chosen an x value within delta of a, then the value of f(x) is within epsilon of the limit L." width="975" height="347" />
Visit the following applet to experiment with finding values of δ δ for selected values of ε:
Example 2.39 shows how you can use this definition to prove a statement about the limit of a specific function at a specified value.
