Calculus/Formal Definition of the Limit

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Formal Definition of the Limit


Whenever a point is within units of , is within units of


In preliminary calculus, the concept of a limit is probably the most difficult one to grasp (after all, it took mathematicians 150 years to arrive at it); it is also the most important and most useful one.

The intuitive definition of a limit is inadequate to prove anything rigorously about it. The problem lies in the vague term "arbitrarily close". We discussed earlier that the meaning of this term is that the closer gets to the specified value, the closer the function must get to the limit, so that however close we want the function to the limit, we can accomplish this by making sufficiently close to our value. We can express this requirement technically as follows:

Definition: (Formal definition of a limit)

Let be a function defined on an open interval that contains , except possibly at . Let be a number. Then we say that

if, for every , there exists a such that for all with

we have

To further explain, earlier we said that "however close we want the function to the limit, we can find a corresponding close to our value." Using our new notation of epsilon () and delta (), we mean that if we want to make within of , the limit, then we know that making within of puts it there.

Again, since this is tricky, let's resume our example from before: , at . To start, let's say we want to be within .01 of the limit. We know by now that the limit should be 4, so we say: for , there is some so that as long as , then .

To show this, we can pick any that is bigger than 0, so long as it works. For example, you might pick , because you are absolutely sure that if is within of 2, then will be within of 4. This works for . But we can't just pick a specific value for , like 0.01, because we said in our definition "for every ." This means that we need to be able to show an infinite number of s, one for each . We can't list an infinite number of s!

Of course, we know of a very good way to do this; we simply create a function, so that for every , it can give us a . In this case, one definition of that works is (see example 5 in choosing delta for an explanation of how this delta was chosen.)

So, in general, how do you show that tends to as tends to  ? Well imagine somebody gave you a small number (e.g., say ). Then you have to find a and show that whenever we have . Now if that person gave you a smaller (say ) then you would have to find another , but this time with 0.03 replaced by 0.002. If you can do this for any choice of then you have shown that tends to as tends to . Of course, the way you would do this in general would be to create a function giving you a for every , just as in the example above.

Formal Definition of the Limit at InfinityEdit

Definition: (Limit of a function at infinity)

We call   the limit of   as   approaches   if for every number   there exists a   such that whenever   we have

 

When this holds we write

 

or

  as  

Similarly, we call   the limit of   as   approaches   if for every number   , there exists a number   such that whenever   we have

 

When this holds we write

 

or

  as  

Notice the difference in these two definitions. For the limit of   as   approaches   we are interested in those   such that   . For the limit of   as   approaches   we are interested in those   such that   .

ExamplesEdit

Here are some examples of the formal definition.

Example 1

We know from earlier in the chapter that

 

What is   when   for this limit?

We start with the desired conclusion and substitute the given values for   and   :

 

Then we solve the inequality for   :

 

This is the same as saying

 

(We want the thing in the middle of the inequality to be   because that's where we're taking the limit.) We normally choose the smaller of   and   for  , so   , but any smaller number will also work.

Example 2

What is the limit of   as   approaches 4?

There are two steps to answering such a question; first we must determine the answer — this is where intuition and guessing is useful, as well as the informal definition of a limit — and then we must prove that the answer is right.

In this case, 11 is the limit because we know   is a continuous function whose domain is all real numbers. Thus, we can find the limit by just substituting 4 in for   , so the answer is   .

We're not done, though, because we never proved any of the limit laws rigorously; we just stated them. In fact, we couldn't have proved them, because we didn't have the formal definition of the limit yet, Therefore, in order to be sure that 11 is the right answer, we need to prove that no matter what value of   is given to us, we can find a value of   such that

 

whenever

 

For this particular problem, letting   works (see choosing delta for help in determining the value of   to use in other problems). Now, we have to prove

 

given that

 

Since   , we know

 

which is what we wished to prove.

Example 3

What is the limit of   as   approaches 4?

As before, we use what we learned earlier in this chapter to guess that the limit is   . Also as before, we pull out of thin air that

 

Note that, since   is always positive, so is   , as required. Now, we have to prove

 

given that

  .

We know that

 

(because of the triangle inequality), so

 
Example 4

Show that the limit of   as   approaches 0 does not exist.

We will proceed by contradiction. Suppose the limit exists; call it   . For simplicity, we'll assume that   ; the case for   is similar. Choose   . Then if the limit were   there would be some   such that   for every   with   . But, for every   , there exists some (possibly very large)   such that   , but   , a contradiction.

Example 5

What is the limit of   as   approaches 0?

By the Squeeze Theorem, we know the answer should be 0. To prove this, we let   . Then for all   , if   , then   as required.

Example 6

Suppose that   and   . What is   ?

Of course, we know the answer should be   , but now we can prove this rigorously. Given some   , we know there's a   such that, for any   with   ,   (since the definition of limit says "for any  ", so it must be true for   as well). Similarly, there's a   such that, for any   with   ,   . We can set   to be the lesser of   and   . Then, for any   with   ,   , as required.

If you like, you can prove the other limit rules too using the new definition. Mathematicians have already done this, which is how we know the rules work. Therefore, when computing a limit from now on, we can go back to just using the rules and still be confident that our limit is correct according to the rigorous definition.

Formal Definition of a Limit Being InfinityEdit

Definition: (Formal definition of a limit being infinity)

Let   be a function defined on an open interval   that contains   , except possibly at   . Then we say that

 

if, for every   , there exists a   such that for all   with

 

we have

  .

When this holds we write

 

or

  as  

Similarly, we say that

 

if, for every   , there exists a   such that for all   with

 

we have

  .

When this holds we write

 

or

  as   .
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Formal Definition of the Limit