Having discussed how you can compute limits, I want to examine the definition of a limit in more detail.

You might wonder why it is necessary to be careful. Suppose you're trying to compute . You might think of drawing a graph; many graphing calculators, for instance, produce a graph like the one below:

It looks as though the graph is dropping down to 0 near . From this, you might guess that the limit is 0. In fact,

It's possible to justify this algebraically once you know a little about limits of trig functions.

Pictures can be helpful; so can experimenting with numbers. In many cases, pictures and numerical experiments are inconclusive or even misleading. In these cases, how can you determine whether a proposed answer is correct or not?

Because the limit definition is a bit abstract, I'll start off with an informal definition.

* Informal Definition.* If can be made arbitrarily close to L for all x's
sufficiently close to c, then

This statement is like a guarantee. Think of making parts in a factory. Your customers won't buy your parts unless they meet certain specifications. So you might guarantee that your parts will be within 0.01 of the customer's specification.

Likewise, to say that you must be able to guarantee that you can make fall within (say) 0.01 of L. But you have to do more:
You have to be able to make fall within *any
positive tolerance* of L --- 0.0001, 0.0000004, and so on, no
matter how small.

Another way to think of this is as *meeting a challenge*; for
example:

* Challenge:* "I challenge you to make stay within 0.0005 of L."

* Your response:* "I guarantee that every x
within 0.003 of c (except perhaps c itself) will give an that is within 0.0005 of L."

To prove that , you
must be able to meet the challenge *no matter what positive
number* is used in place of 0.0005.

By the way, notice that is excluded in my guarantee.
The reason is that in computing , we're concerned with what happens as x
*approaches* c, not what is.

Before I give some examples, here's an important fact about absolute value:

We want absolute values, which are always *nonnegative*,
because a distance shouldn't be negative.

Also, notice that

That makes sense, because the distance from A to B should be the same as the distance from B to A. For instance,

* Example.* By plugging in , it appears that

How close should x be to 4 to guarantee that is within 0.01 of 7?

Let's work backwards: I want to be within 0.01 of 7. This means

The last inequality says that the distance from x to 4 should be less than . So if x lies within of 4, I can guarantee that will be within 0.01 of 7.

A formal proof would just reverse the steps above:

Can you see that if I'm challenged to make lie within 0.00001 of 7, I should make x lie within of 4? Just replace 0.01 with 0.00001 in the discussion above.

And similarly, I can make lie within any tolerance FOO of 7 by making x lie within of 4.

This shows that I can meet *any challenge*, since I can just
take the challenge tolerance and plug it in for FOO. This proves that

* Example.* The graph of a function is shown below.

It appears that .

A (grey) horizontal strip of width 0.5 is drawn around . Draw a picture to show a range of x-values around 4 for which the corresponding -values lie in the horizontal strip.

Use it to estimate the width of a symmetric vertical strip around 4 representing x-values whose corresponding -values lie in the horizontal strip.

Suppose I'm challenged to make fall within 0.5 of 3. That is, I want my y-values to fall within the grey strip in the picture.

On the right side of 4, the graph stays within the grey strip as far as 4.25; on the left side of 4, the graph stays within the grey strip as far as 3.

If I want a strip that's symmetric about 4, I use the closer of the two values, which is 4.25. Now 4.25 is 0.25 units from 4, so my answer is: If x is within 0.25 of 4, then will be within 0.5 of 3.

If I can meet such a challenge with *any positive number* in
place of 0.5, then I will have proved that .

* Example.* (Disproving a limit) Consider the
function whose graph is show below.

Suppose that Calvin Butterball thinks that . Use the limit definition to disprove it.

To disprove Calvin's claim, I'll make a challenge that Calvin can't meet.

I challenge Calvin to make fall within 0.5 of 4. This means that he must find a range of x's around 3 so that the corresponding part of the graph lies within the grey strip shown below:

You can see that there's no way to do this. (Note: He's not allowed
to use alone. Remember that what the function does
*at* has no bearing on the value of the limit.)

Since this challenge can't be met, . In fact, is *undefined*.

* Example.* Suppose

It appears that . How close should x be to 1 in order to guarantee that will be within 0.0008 of 3?

As in an earlier example, I'll work backwards.

From the left side, I'd need

The last inequality says that x should be within 0.0004 of 1.

From the right side, I'd need

This means that x should be within 0.0002 of 1.

To satisfy the two requirements at the same time, I'll use the smaller of the two numbers. So I'll require that x should be within 0.0002 of 1, which means

Here is the "real" proof, which I get by writing the scratch work in the reverse order.

Suppose . If , I have

Now

So if , I have

(From the second to the third line, I used the fact that .)

Thus, if x is within 0.0002 of 1, then will be within 0.0008 of 3.

I'm almost ready to give the formal definition of a limit, but I need to mention something first as a matter of honesty. It's a technical issue, and it won't arise in the majority of problems and examples (so you can ignore it without much harm if you wish).

* A technical point.* In discussing , I'll usually assume that
f is defined on an open interval containing c. That is, there are
numbers a and b such that and f is defined (at
least) on .

For one-sided limit (which I'll discuss later), should be defined on an open interval with c as an endpoint.

To understand why you want to do this, consider the function

(So, for instance, f is simply not defined at , or at .)

In the definition of , the "if" part of the definition would
hold vacuously (for small open intervals around -10), because there
would be *no values* of x near -10 for which f was defined.
Thus, the limit L could be anything!

The condition on the domain of f is made to avoid silly cases like this one.

In order to avoid cluttering the statements of the definition or of proofs of limit properties, I usually won't state this assumption about the domains of functions in limits explicitly.

Now I'll give the formal definition of a limit, and show how to use it to do proofs.

* Definition.* means:

For every , there is a , such that for all x in the domain of f, if , then .

" " is the Greek letter epsilon. It is the "challenge number", the tolerance or maximum error you have to meet. is the Greek letter delta. It is the "response number", the setting on x which meets the challenge. The Greek letters are used in this definition for traditional reaons; there is nothing otherwise special about them.

Let's see how proofs of limits work using the definition.

* Example.* Prove that .

In this problem, 4 corresponds to c, corresponds to , and 25 corresponds to L in the limit definition.

I have to show that, given any , there is a , such that

Notice that I'm *given* , but I'm not told
its value (which was the case in earlier examples). All I can assume
is that it's some positive number. I have to come up with a that meets the condition above. To do this, I work
backwards as I did in earlier examples. This is
"scratchwork", and doesn't count as the "real"
proof, which will come afterward.

* Scratchwork.* I want . I'll work backwards from this
and try to get something that looks like " ". Then I'll set and try to do the real proof.

Okay --- I'll try .

* The real proof.* Let . I must show that

When you are proving an "if-then" statement, you get to
*assume* the "if" part, and you *prove* the
"then" part. So assume

The rest of the proof is easy: I just reverse the steps I did on scratchwork:

Therefore, by the limit definition,

A similar approach works for limits of the form . Here is a harder example.

* Example.* Prove that .

In this case, 3 corresponds to c, corresponds to , and 24 corresponds to L.

* Scratchwork.* I want . I'll work backwards from this
and try to get something that looks like " ". Then I'll set and try to do the real proof.

I can't just divide both sides by (like I divided by 7 in the last example:

The problem is that I can't set , because I would need to know x in order to know --- but is supposed to determine the range of x's.

Instead, I need to make a "preliminary" setting of . I'll provisionally set . Then

Remember that you have *complete control* over . Setting to 1 is like
adjusting a setting on an instrument, where you make an initial rough
setting, then fine-tune it. We'll see how this works out when we
write the "real proof".

Adding 8 to each term, I get

Remember that I want the inequality .

If I could get , then I'd have

But

It looks like I should try ... but then, I remember I needed to set earlier. How can I get *both* of these things
to happen? The idea is to make the
*smaller* of the two numbers 1 and --- in symbols,

("min" stands for "minimum".) This means that

* The real proof.* Let . I
must show that:

So I may *assume* ,
and I have to *prove* .

As I noted in the scratchwork, I know that

Take first. Then

Next, I'll use . Multiply this inequality and the inequality to get

Therefore, I've proved that .

Copyright 2018 by Bruce Ikenaga