In this video, I will illustrate with an example
how to prove that a function does not have a limit;
that the limit does not exist
directly from the formal definition.
I assume you have already watched the videos where I explain the definition
and the examples
on how to prove that a function DOES have a limit from the definition.
I will include links in the description.
I could say that the example in this video
is like the previous ones on steroids.
Okay, let's get started.
I will work with the function h,
defined by the equation h(x) = cos(π ⋅ x).
I want to prove that lim_(x → ∞) h(x) does not exist.
To understand this function, first let’s look at its graph.
It oscillates between -1 and 1.
And for large values of x, as x approaches infinity,
it keeps oscillating.
Geometrically, I can see why it does not have a limit at infinity,
as it does not approach one single number.
The π in the argument of cosine makes calculations easier.
When x is an EVEN integer,
I get cosine of 2π, 4π, 6π, and so on, which is always 1.
When x is an ODD integer,
I get cosine of π, 3π, 5π, and so on, which is always -1.
This will come in handy.
Okay, then.
Can we write a rigorous proof?
If I want to prove that the limit does not exist,
then I need the formal definition of “the limit does not exist”.
Let’s figure it out.
As a starting point,
we know what it means that lim_(x → ∞) h(x) is a number L;
it is THIS statement.
If you are NOT comfortable with this definition,
watch the video where I came up with it. I will link it in the description.
To say the limit is NOT L, I need to negate the previous statement.
Let’s do that.
I need to say it is FALSE that every ε satisfies something.
In other words, there exists ONE ε that fails the rest of the sentence.
Now, I need to say it is FALSE
that there is a real number M that satisfies the implication.
This means that EVERY real number M fails the implication.
And now it gets a bit tricky.
When I write the implication, I mean that EVERY real number x satisfies it.
There is a hidden quantifier, "for all x," in here.
So to negate the implication,
I need to say that AT LEAST ONE REAL NUMBER x fails it.
And finally, to say the implication is FALSE,
I need to say that the "IF" part is TRUE and the "THEN" part is FALSE.
That’s it.
If the last part confused you, I suggest you rewatch the video
where I explain, more slowly, how to negate a conditional,
and why there is sometimes a hidden quantifier.
I will link it in the description.
But we are not done.
What I have written is that the limit does not equal some number L,
for example, that the limit is not 5.
Saying that the limit DOES NOT EXIST means that it is not ANY real number.
In other words, FOR EVERY REAL NUMBER L, the limit does not equal L.
Since I already have a definition of “the limit does not equal L”,
I can append it here...
and that’s it!
This final statement, now in blue at the bottom,
is the definition I was looking for.
Four quantifiers, no less.
At this moment,
I suggest you pause the video to convince yourself
that you understand this statement and where it came from.
This is a long, scary-looking statement.
But I can treat it like any other statement I want to prove.
Before going further,
I will look at it carefully to figure out
what the structure of the proof should be.
Okay. THIS is what I want to show.
So my proof MUST begin by FIXING an arbitrary real number L.
THEN, I will need to SAY what I take as ε,
possibly as a function of L.
THEN, I will FIX an arbitrary real number M.
THEN, I will need to SAY what I take as x,
possibly as a function of L and M.
And finally, I need to verify the two inequalities.
I went a bit faster, because it is not the first time
that I write the structure of a proof like this.
If it went too fast, you can rewatch some easier epsilon-delta proofs,
linked in the description.
The difficulty, as usual,
is to figure out what to take as ε and as x.
Let’s explore it.
I'm going to look at the WHOLE statement against a graph
to try to understand what it means better,
and to figure out what I pick as ε and what I pick as x.
We begin with "for every L," so L is an arbitrary real number there.
Now, I need to say what ε is,
so I'll be picking a value of ε, for example, up to here.
this is L + ε, and this is L - ε.
And the last condition here
means that h(x) will NOT be in this interval;
will NOT be between L + ε and L - ε.
So I'll be picking probably a part in here.
Now, "for EVERY real number M."
So let's say M is here,
and I need to show that there exists ONE value of x
which is greater than M, and with h(x) outside of that interval.
So for example, I could pick x to be here,
and it will give me h(x) outside of the interval.
So, these two conditions mean that when I pick the point x,
h(x) doesn't land here.
So I need to avoid this region.
As long as I can find a point to the right of M
that avoids this region,
I am fine.
The question was,
what do we pick as ε, and what do I pick as x?
It should be clear that to make this happen,
I'm gonna need to pick ε not too large;
if I pick a value of ε that is very large
so that (L + ε, L - ε) includes everything between -1 and 1,
this will be impossible.
So as long as I pick ε small enough
so that it doesn't include EVERYTHING between -1 and 1,
I have a hope of making this work.
So, I am going to pick ANY ε which is at most 1,
because then, either 1 or -1 -- at least one of them --
has to be missing from the interval (L + ε, L - ε),
because in order for the OPEN interval to include BOTH -1 and 1,
the diameter would have to be STRICTLY greater than 2,
so ε would have to be STRICTLY greater than 1.
And then, which value of x do I pick?
I will pick an integer x.
Then, h(x) will be either 1 or -1, depending on whether x is odd or even.
This will satisfy what I need; h(x) WON'T be between L + ε and L - ε.
I also need to pick x > M.
But there are certainly integers as large as I want.
So I think that will work.
And I think that's all I need!
I think I am ready to write a proof now!
Here is, once again, the claim I have to prove.
And, at last, here is the actual proof.
First, I FIX an arbitrary real number L.
Then I PICK ε to be 1/2.
How do I know to pick THIS ε?
Because of all the preparation work I did earlier!
I know that I could have chosen ε depending on L if needed,
but it turns out I do not need to.
Then, I FIX an arbitrary real number M.
AT LEAST one of the following must be true:
Either 1 or -1
is not in the open interval from L - ε to L + ε,
thanks to my choice of ε.
In the first case,
I will choose any integer x that is EVEN and greater than M.
There certainly is such an integer x.
If necessary, I could write an explicit formula for it,
probably using the floor function or the ceiling function.
But I do not think
that it would make the proof any more clear or convincing,
so I will just keep it this way.
And with this choice, h(x) is 1.
In the second case,
I choose any integer x that is ODD and greater than M.
And then, h(x) is -1.
Either way, x is greater than M,
and the distance between h(x) and L is greater than or equal to ε,
which is what I wanted to prove.
We are done. Hooray!
One final observation:
Look at the parts of the proof that I just coloured in blue.
The structure of the proof
is exactly what we agreed earlier that it should be.
This is always a good thing to check.