In this video, I will introduce the FORMAL definition of limit.
I will assume that, before this one, you have watched
the three videos in which I explain
the intuitive idea of limit geometrically,
and the video where I talk about absolute values and distance.
Let's begin.
This is my starting point.
When I say lim_(x → a) f(x) = L,
I'm thinking of a function like the one in this graph.
Perhaps it's not even defined at a and the graph has a hole at a,
but it's "nice" when I'm approaching a.
Roughly, the idea I had
was that IF x is close to a but NOT a, THEN f(x) is close to L.
And I want to take this rough idea,
and rewrite it into something formal that has a very precise meaning.
I don't care what happens at a; perhaps it has a hole,
perhaps it's defined with the "wrong" value,
perhaps it doesn't have a hole at all.
I'm going to treat all those the same way,
and I'm going to be thinking of something like the one in the picture.
So let me take this rough idea,
and let's examine it part by part, and try to formalize it.
First, how do I write that x is close to a?
Well, I can say that the distance between x and a is "small,"
and according to the video on absolute values,
that's the same thing as saying "the absolute value of x - a is 'small.'"
But "small" is not a precise term. So instead, I am going to say,
"the absolute value of x - a is smaller than delta (δ)."
I am choosing the Greek letter delta (δ) here.
I could have chosen any other letter. It's traditional to use delta (δ).
And δ is going to be some kind of measurement;
some kind of cut-off that tells me what "small" means.
I will have to explain what that means later.
But for now, if I choose δ to be some small number,
"the absolute value of x - a is smaller than THAT"
means that they are close.
I also need to say that x is NOT equal to a.
What I'm going to do is add a second inequality and say
that the absolute value of x - a is GREATER THAN 0.
Because the ONLY way the absolute value could be 0
is when x = a.
So that is there simply to exclude the case x = a.
And now, the statement at the top is a conditional.
So IF x is close to a but not a
-- that means IF the distance between x and a is between 0 and δ --
THEN f(x) is close to L.
So I'm going to say that the distance between f(x) and L
is "small"; smaller than epsilon (ε);
I am going to choose a SECOND cut-off;
a second way to measure what it means to be "small."
This implication is going to be a piece in the definition of limit,
but now I need to think hard and figure out:
what do I actually mean by δ and by ε?
I have some idea
that they should be small numbers so that they represent the idea
that x is close to a and f(x) is close to L.
But exactly what, I still need to figure out.
Next, I will put this in the context of a properly written definition.
I'm going to say
that lim_(x → a) f(x) = L when this implication is true.
But as I've written it, this is neither right nor wrong;
this is meaningless
because there are so many letters that I haven't defined.
I need to explain what ALL the pieces of notation mean.
I'll take care of the easier ones first.
L and a are two real numbers
that I need to fix before the definition even starts.
So let a and L be real numbers.
And f should be a function.
Now, in order to study this limit, I don't care if f(a) is defined or not.
But f needs to be defined for values of x CLOSE to a.
So I'm going to say that f is a function
defined AT LEAST on an interval centered at a, except maybe at a;
it may or may not be defined at a.
This is better.
Now, I still need to explain what I'm doing with ε and δ;
let's look at those.
I'm going to look at a few examples
of graphs of functions that do or do not have a limit
to understand what the implication means geometrically,
and what conditions I need to ask about δ and ε.
The "IF" part says
that the distance between x and a is between 0 and δ, exclusive.
So x must be in this interval, but NOT be a.
And the "THEN" part says
that the distance between f(x) and L is smaller than ε.
So f(x) is in this interval.
So whenever x is in this interval but not a,
f(x) is in this interval
In other words, I'm in this region,
and these two other regions are forbidden.
As long as they avoid these two regions, I will satisfy the implication.
Notice that for other values of x,
there are no constraints and f(x) can be anything.
For example, the function with THIS graph satisfies the implication
for the specific value of ε and the specific value of δ
that I chose in the picture.
I want to figure out what I need to say about δ and ε.
I hope it is clear that we are NOT asking
for the implication to be true for ALL values of ε and ALL values of δ;
there are pairs of values for which this won't work.
But it is also not enough to request the implication
for just one value of δ and one value of ε.
Let's look at an example of a function that doesn't have a limit for comparison.
Here is the graph of a function that does not have a limit at a.
So, this example should FAIL the definition.
Now for the specific value of δ
and the specific value of ε in the picture,
the function SATISFIES the implication,
because the graph AVOIDS the forbidden regions.
So which values of ε and δ will notice that this function doesn't have a limit?
I have an idea that I want "small" δ and "small" ε,
because the distances should be small.
Now, making δ small is not going to help; look at it.
No matter how small I make δ,
this function is going to continue satisfying the implication.
In fact, making δ smaller makes it EASIER to satisfy the implication,
because it NARROWS the forbidden region.
On the other hand, if I make ε small; if I make it small enough,
then eventually I will notice that this function fails to have a limit.
So in order to capture
that this function doesn't have a limit and the other one does,
I need to look at VALUES of ε specifically small.
I'm going to request that this is true for ALL values of ε positive.
And when I say for "all," I mean "small."
I kind of wish I could say, "for the smallest positive value of ε."
But there IS NO smallest positive value of ε;
I cannot make ε 0, and there's no smallest positive real.
So that's why I say
that I want the implication to be true for all ε.
But in my mind, I think that I'm going to take values of ε small
and smaller and smaller and as small as possible,
and see if those still work.
And then, for every value of ε, I don't need lots of values of δ;
I just need ONE single value of δ for each value of ε.
So I'm going to request that for every value of ε,
there exists one value of δ that makes the implication true.
Let's see what this means.
In this example; in the example that HAS a limit,
for THIS value of ε, there IS one value of δ that works.
If I make ε smaller, I will have to choose δ,
but there will be another value of δ, also smaller, that works.
And if I make ε EVEN smaller,
well, I can find a value of δ that's EVEN smaller that works.
And no matter how small ε is,
I can find a value of δ that makes the implication work.
By contrast, let's look at the bad example:
the function without a limit.
In this case, for THIS value of ε, there is a value of δ that works.
If I take a smaller value of ε,
maybe there is still a smaller value of δ that works.
But if I take a value of ε that's very small,
so that the interval is only THIS,
then NO value of δ works;
for THAT value of ε, ALL values of δ fail.
Great! This definition is going to realize
that this function DOESN'T have a limit, and the previous one HAS a limit.
There we have it! This is the formal definition of limit.
Hooray!
It looks a bit complex, but unfortunately there is no way to simplify it.
If you omit a little part; if you modify it a little,
chances are, you will drastically change the meaning
and it will no longer mean what we understand as limit.
I recommend that you don't memorize this as a string of symbols,
but that you make sure you understand
why every little piece of the definition is necessary;
that's a MUCH better way to memorize it.
I need to add one very important comment, though:
The second line is an implication; an if-then statement:
IF the distance between x and a is between 0 and δ,
THEN the distance between f(x) and L is smaller than ε.
And when I write an if-then statement like this,
there is a HIDDEN IMPLIED quantifier.
When I write this conditional,
I'm saying this conditional is true for EVERY real value x.
So we could as well just put in there "for every x in the reals."
And we probably should;
probably would make things simpler or more clear.
But unfortunately, it is a standard convention to assume
that this "for all x in the reals" is implied and not write it explicitly.
Choose whatever you want;
if it's going to make you less confused, put it explicitly;
if you're comfortable, don't put it.
But remember that it's there, even if only implied.
To conclude, I will leave you with an exercise.
The BEST way to make sure that you understood this definition
is see if you can generalize it.
In previous videos, I've introduced other notions
like limit on the left, limit on the right,
or a limit being infinity.
If you understood this video, now you can take THOSE geometric ideas
and try to REWRITE them as rigorous definitions.
And if you succeed,
it means you understand the formal definition of limit well.