One of the most important topics in elementary calculus is the definition of limits. The definition says that the if and only if, for all , there exists a such that if , then . In this article, we are going to discuss what this definition means. Readers of this article must have knowledge about elementary calculus and the concept of limits.
Review of Limit Basics
Consider the function . We have learned from elementary calculus that . Aside from algebraic computation, this is evident from the color-coded graph and the table shown in Figure 1. The yellow arrows in the graph and the values in the yellow cells in the table indicate that as the value of approaches from the left of the x-axis, the value of approaches from below of the y-axis. On the other hand, the red arrows in the graph and the values in the red cells in the table indicate that as the value of approaches on from the right of the x-axis, the value of approaches from above of the y-axis.
From the above discussion, it is noteworthy to mention three things:
- We can get as close to as we please by choosing an sufficiently close to . For example, I can set to (with nines) to get an very close to , which is ( nines).
- No matter how small is the distance of from , a distance less than it may still be chosen. For example, if we choose the point which is very close to , say a point with coordinate with ( nines), we can still choose a value closer than this to . For instance, we can choose with nines. This can be repeated for every chosen distance.
- Although can be very very close to , it does not necessarily mean that equals .
Now we go back to the definition of limits. In a specific example, the limit definition states that the if (and only if) for all distance (denoted by the Greek letter ) from along the y-axis (directly above or below ) – no matter how small – we can always find a certain distance (denoted by ) from along the x-axis (left or right of ) such that if is between and , then would lie between and .
To give you a more concrete example, suppose we want the distance from , which is our limit, to be then the interval of our is (. The definition of limit says that given a distance , we can find a distance in the x-axis such that if is between and , we are sure that is between and . We do not know the value of yet, but we will calculate it later.
In Figure 2, is between and or . Subtracting from all terms of the inequality, we have . If you recall the definition of absolute value, this is precisely the same as . The comparison among the notations is in Table 1.
Using the notations in the table, we can conclude that the following statements are equivalent:
- Words: Given , we can find a such that if is between , then is between and .
- Set Notation: Given , we can find a such that if , then .
- Relational Operator: Given , we can find a such that if , then .
- Absolute Value: Given , we can find a such that if , then .
We have discussed that we can get as close to as we please
by choosing an sufficiently close to . This is equivalent to choosing an extremely small , no matter how small, as long as . Our next task is to find the that corresponds to that .
Applying this definition to our example, we can say the if and only if, given (any small distance above and below 4), we can find a (any distance from x to the left and right of ) such that if , then .
The Definition of a Limit of a Function
Now, notice that is the limit of the function as approaches . If we let the limit of a function be equal to and be the fixed value that approaches, then we can say that if and only if, for any (any small distance above and below ), we can find a (any small distance from to the left and to the right of a) such that if then, . And that is precisely, the definition of limits that we have stated in the first paragraph of this article.
In mathematics, the phrase “for any” is the same as “for all” and is denoted by the symbol . In addition, the phrase “we can find” is also the same as “there exists” and is denoted by the symbol . So, rephrasing the definition above, we have if and only if, , such that if then, . A much shorter version of this definition is the phrase , such that . The symbol stands for if and only if and the symbol is similiar to if-then. If and are statements, the statement is the same as the statement of the form “If then “.
Finding a specific delta
We said that given any positive , we can find a specific , no matter how small our is. So let us try our first specific value .
From the definition, we have if and only if, given (any small distance above and below 4), such that if then, .
Now . This implies that
which implies that . Simplifying, we have . This means that our should be between and to be sure that our is between and . This is shown in Figure 4.
Now, let . This means that our interval is . Now . Thus, which implies that . Solving, we have . This means that our should be between and to be sure that our is between and . There are only two examples above, but the definition tells us that we can choose any so let us generalize our statement by doing so.
Now . This results to which implies that . Solving, we have . From the condition above, so we can let .
This means given any , we just let our equal to and we are sure that if is between and to be sure that our is between and .
In the next calculus post, we are going to discuss the strategies on how to get given an arbitrary value, so keep posted.
- Understanding epsilon-delta proofs
- An Intituive Introduction to Limits
- Counting the Infinite: A Glimpse at Infinite Sets
- Epsilon-delta proof: example 1