PROFESSOR: OK. I promised a video about
limits and continuous functions and here it is. So I'll begin with the most
basic idea and with a picture instead of definition in symbols
first. So the most basic idea is that I have
a bunch of numbers-- let's make them positive
numbers-- and I want to know, what
does it mean for them to approach a limit-- capital A-- as I go out this sequence of
numbers, a1, a2, a3, a4. And let me say right away, the
first four numbers, the first million numbers, make no
difference about the limit. So here's what it means. For example, there's a
equals 7, let's say. What does it mean for these
numbers to approach 7? It means that if I take any thin
little space around a, above and below, the numbers
can start out whatever. They can go in there, they could
go out, they could come back, whatever, they could
grow way big, way small. But in the end, beyond some
point, eventually, they have to get in that slit
and stay in there. And the slit, then,
could be smaller. And then they would have to get
into that smaller slit and stay there. So that's what it means for
the numbers to approach A, that eventually after any number
of jogs around, they get inside and they stay
there, however thin that slit is. A slight difference when
a is 0, because the numbers are positive. They're coming down,
they get in. And again, they must stay in. And then, again, I'm going to
make the band tighter, and they have to get into
that and stay there. And what does it mean for the
numbers to approach infinity? That means that whatever-- so this is often
called epsilon. I'll use that Greek
letter epsilon as a very small number. So this would be A minus
epsilon, and this would be A plus epsilon. And then the epsilon could
be made smaller. And now here is some
big number, like even 1 over epsilon. So that's a giant number. And the limit is infinity if,
again, they can dodge around for a while, they can go
down, they can go up. But eventually, they
must get above that line and stay there. And if I move the line up
further, they have to get above that line for me to
say that the limit is-- so I have these possible
limits. Infinite, some positive,
ordinary number, and 0. Those are possible limits. But of course many sequences
have no limit at all, like sine of n, it will just bounce
around, cosine n-- many, many things. OK. So I think that the way to get
the idea, use the idea, is to ask some questions
about limits. And we'll see that usually the
answer is yes, OK, no problem. But once in a while, for certain
limits are dangerous. So really always mathematicians
are looking for what's special, what unusual
thing could happen? Because the truth is, limits are
ordinarily rather boring. If the an's approach 7 and the
bn's approach 4, so the a's get close to 7 and the b's get
close to 4, then their differences will get
close to 7 minus 4. But is there any case in
which that could fail? Is there any case among these
in which we could not know what the limit was, and it might
not exist, or it might be like any number? And I think that can happen
in this, so I've got four different questions
here, getting more interesting as we go down. In the first one, I can
see only one problem. If the a's approach infinity, so
they get very big, and the b's also approach infinity, get
very big, so capital A and capital B become formally
infinity minus infinity, and we don't know the
answer there. That has no meaning. So this'll be my little
list of danger. I mean, it's not like
skydiving, but for a mathematician this
is high risk. OK, so how could this happen? Well, the an might
be n squared. And the bn might be only n. Right? So they're both going to
infinity, n squared and n. But n squared is going faster. It's like a race. n squared will win, and the
difference between them will actually grow faster
and faster. Or they could go to
infinity together. an and bn could both be n,
both headed for infinity. The differences would be
0 all along, n minus n. So the limit of the difference
would be 0 minus 0. So this could be 0, but it could
be infinity, it could be minus infinity, it could
be anything. Any limit is possible there. Do you see that there
is a case-- it's sort of a special case,
because it only happens when these limits are infinite-- but now it's sort of OK
to look at each-- let me look at number two. How about multiplication? If I multiply a bunch of numbers
that are headed for 7 and a bunch of numbers that
are headed for 4, their product is going
to head for 28. This will be true. When could it fail? Well, again, it's going to be
extreme cases, because if I have ordinary numbers for
A and B like 7 and 4, there's no doubt. But look at the extreme
case of when A is 0 and B is infinite. So the an's are headed for 0. The bn's are getting bigger
and bigger, the an's are getting small as we
go far enough out. OK. In that case, well,
again it's a race. The an's might be 1 over
n squared, and the bn's might be n. So this would be n over
n squared, and that would go to 0. But if I reverse those I could
have n squared times 1 over n. The product could get bigger,
or the product could-- all possibilities. All possibilities there. So I cannot know what
that one is. 0 times infinity
is meaningless. OK. What about number three? The danger increases as soon
as we start dividing. I made the b's positive,
but I don't know if capital B is positive. So the danger-- and, in fact, the most important
case for calculus-- is 0 over 0. If the a's go to 0 and the b's
go to 0, I can't tell what their ratio goes to, because it
depends how fast they go. If the a's go quickly to 0 and
the b's are rather slow getting there-- in other words the b's would be
a lot bigger than the a's even though both are
going to 0-- then that fraction
would be small. But if I reverse them, the
fraction would be large. So I think 0 over
0 is a danger. I think there's another
danger here. Yeah, maybe infinity
over infinity. Again, that's a race that we
can't tell, until we know details about the sequences,
who's going to win. If they an's go off to infinity
and the bn's go off to infinity, ah, a very
important case. The ratio could be
1 all along. The a's and b's could be the
same, headed for infinity. Or the an's might be squaring
the b's and going up faster, or the square root of the
b's and going slower. So again, infinity over
infinity, we can't-- 0 over infinity, if the a's are
headed for 0 and the b's are headed big, then that ratio
is going to be small and head for 0. 0 over infinity, I'm OK with. Call it 0. Well, I don't know if
that's legal, but anyway, let me do it. All right, last one of
this kind, just for practice about limits. Again, you see what I'm
constantly doing is thinking of examples that simply
show that I can't be sure of the limit. So here normally I could be
sure, if this is headed for 7 to the fourth, that'll be the
limit, whatever that is, 49 squared, 2401, or something. But if-- now when could it go wrong? Here's an interesting case. So this is my list of danger,
and I think I'm in danger if they both go to 0. 0 to the 0-th power, I don't
know what that is. And actually, I don't know all
the possibilities here. I can see one way would be let's
suppose the b's were actually 0, or practically. Then things to the
0 power are 1. So I could get the answer 1 here
by fixing the b's at 0 and letting these guys, they
would all be to the 0 power, so they would all be 1. And in the limit,
I would have 1. But I could also do
it differently. I could fix these at 0 and let
these guys get smaller. Then I would have 0 to powers. And zero to any power is 0. You see my little
problem here? Let me write my little
problem here. My problem is that a to the 0
power would be 1, but 0 to the a-th power would be 0, or 0 to
the b-th, maybe I should say. So if I'm in this situation and
the a is shrinking to 0, I still have a limit of 1's. But if I'm in this situation and
the b's are headed for 0, I have a limit of 0's. And maybe you could get
1/2, I don't know how. And you have to allow me--
because I have to finish this list, and I only have one
more to tell you-- that another case, a very
interesting type of calculus case is the case where
the a's go to 1 and the b's go to infinity. I don't know if you remember
that this actually happened in the lecture on e, the number
that comes in e to the x, the great number of calculus. Do you remember that? So I'm going to talk a little
bit about the a's going to 1 and the b's blowing up. So I'm getting things that are
very near 1, but I'm taking many, many more of them. And I believe that I can
get all kinds of different limits there. I believe I can get all kinds
of different limits. Do you just-- maybe on this next board. And then I promise to come
back to the heart of the subject of limits and continuous
functions. But I just think that one, the
famous case of this one, was 1 plus 1 over n. That's the a's, and that
approaches what limit? One. The b's I'm going
to take as n. So the b's are going
to infinity. So I'm discussing
this case here. So that's a case where
this goes to 1, this goes to infinity. I had an email this week saying,
wait a minute, I've got a little problem here,
because I know 1 to the infinity is e. 1 to the infinity is 3. Well, that's because it's true
that that number approaches e. That's one of the many
remarkable ways to produce the number e, the 2.7-something. But that's because the race
between this and this was so evenly balanced. If I took these closer and
closer to 1, like n squared, what would happen then? Then I'm taking numbers very,
very near 1, I'm taking a power, but these are sort of
near, those would approach-- would you like to guess? One. These are so close to 1 that
taking the nth power doesn't move them far. And you can guess that I could
get infinity too, by taking n not still close to 1
and taking some big power like n squared. Now I have things close to 1,
but I'm taking so many of them that it would blow up. So again i think--So those are
all cases where in the limit, I have 1, in the limit,
I have infinity. But that combination 1 to the
increasingly high powers can do different things. This was my little idea to
show you the risky cases. OK. But actually, 0 over 0, that's
what calculus is always doing, right? Because that's exactly what we
have when we have a delta f over a delta x, a delta
y over a delta x. They're both approaching 0 and
we get a definite slope when the ratio goes to
a good number. OK. So can I discuss 0 over 0? All right, phooey on this one. OK. So I now want to speak
about the case when f of x goes to 0. Let's say f of x goes
to 0 as x goes to 0. So there'll be an if here. I have to say what that means. And then I'm also going to have
some g of x going to 0 as x goes to 0. OK, so both functions
are decreasing. And my question, let me ask the
question first, what about f of x over g of x? What does that do? And of course, just as I said
up there, I can't tell yet. I have to know the
f and the g. It's a race to 0, and I have to
know who's the winner and by how much. But first, I'd better say what
does it mean for a function to go to 0 as x goes to 0. Well, you know. Let me draw a graph
of this function. OK, I'll just draw it. So f of x is going to 0. So here is x, and I'm going to
graph f of x, and here is 0. So as x is coming down to 0, my
f of x is also coming to 0. So it could come like so. That's a pretty sensible,
smooth, nice approach to 0. That could be my f of x. And it may be a g of x is
smaller, but also approaching 0 in a nice, smooth way. This is a case where
you can see those, as x goes that way-- maybe the arrow should be going
that way, because x is going to 0-- my f of x is getting smaller, my
g of x is getting smaller. And I'll say exactly what
that means, but you know what it means. It means that if I put a little,
like these lines, if I put a little band there,
it gets into that band. Actually, g will get into
the band sooner. But then f will safely
get into the band. Now, the question is what about
f of x over g of x? OK. Can we say? Now, I'm going to suppose
that f of x has a definite slope, s. And this one has a definite
slope, t. In other words, I am
going to suppose-- Here look, this is called,
named after a French guy, L'Hopital, the hospital rule. OK, so it's just a little trick,
because this comes up of what's happening
in this race to 0. And the natural idea is that f
of x is really, since f is 0 there, and I'm really just
going a little way. So maybe I call that delta x,
just to emphasize that I'm looking really near 0. And that f of x is really
going to be delta f. And that g of x is really going
to be delta g, because let me draw the picture,
delta f is that height. Here is delta x, and
here is the height. It's because that
point is 0, 0. So the differences I'm taking,
the f of x in the delta, the f of x plus delta x is just f at
delta x, just that height. And g is this smaller one. Do you have an idea of what
this answer's going to be? If I look at that ratio of this function to this function-- here the ratio, I don't know
what, 3 or something. Here it's, I don't know,
maybe 4, maybe more. As I'm getting closer and
closer, this height is controlled by the slope. And this height, the g of x,
is controlled by its slope. Look, here is the
way to see it. Just divide top and
bottom by delta x. Same thing. So I haven't changed
anything yet. I divided the top and the
bottom by delta x, just because now I'll let everything
go to 0, delta x will go to 0, the delta f will
go to 0, so the delta g will go to 0. But I know what this
approaches. Delta f over delta x approaches
the slope, s. And delta g over delta x
approaches the other slope, t. So you see, this is L'Hopital's
rule, that if f goes to 0, and if g goes to 0,
and if they have nice slopes, then the ratio of f to g, which
looks like 0 over 0, we can actually tell what it is by
looking at the derivative, by looking at those slopes. It's the ratio of the slopes. OK, that takes a little
thought and, of course, some practices. It also takes some examples to
show what else could happen. Can I just draw another f,
and you tell me what about f over g. I'm sorry to give you all these
questions, but it's example, answer, that you
get the hang of slopes. Suppose f goes much steeper. I mean, f could be the
square root of x. There's f equal the
square root of x. Square root of x has an
infinite slope at 0. It's a good function to know,
the square root of x, because this is x to the 1/2 power. And its derivative, its slope,
we know will be 1/2 x to the minus 1/2 power. And then as x goes to
0, that blows up the way the picture shows. Now, what would f over g, so
this is a case where f hasn't got a slope. The slope is infinite now.
s is now infinite. And that ratio is going
to blow up. This one is getting
to 0 but slowly. This f is staying much bigger
than the g, and the ratio would be infinite. So there's a case where
L'Hopital can't help because f, this slope s, which was fine
for this nice function, is not fine for this function. The slope is infinite for that
square root function. OK, a bunch of examples that
begin to show what can happen and the need, really, for a
little bit of care on what does it mean? What would I say about that
square root function? So I'll even write
that down here. f of x equals square root
of x at x equals 0. What would I say about
that function that we know it's picture? I would say it has
infinite slope. Or if you prefer, its slope
is not defined. We don't have a good number
there for its slope. But I would still say the
function is continuous because the darn thing does get
below any band. If I draw a little band here,
the function does get into that band and stay inside. It just took a long time. It stayed out of that band as
long as it could and then finally fell in just
at the last minute. OK, so I would say this function
has the slope not defined, not OK at x equals 0. But f of x is continuous
at x equals 0. So I'm trying to make the
distinction between asking for the function to be continuous
is not asking as much. If a function's got a nice
slope, like g, that function's got to be continuous. And more, it has to have
this good slope. This f of x, this square root
function will be continuous. And now I have to tell you
what continuous means. It's not asking for so much as
a slope, because the slope could come down infinitely
at the last minute. All right, so what's a
continuous function? Continuous function means-- a continuous function, f of x,
at some point-- maybe here it was 0, I'd better allow
any old point. So in words, it means f of
x approaches f of a as x approaches a. That's what it means to be
continuous at that point. It means that there is
a number, a value for f at that point. And we approach that value as
we get near that point. That seems such a
natural idea. That's what it means for a
function to be continuous. And with this piece of chalk
or with your pen, it means that I can draw the function
without lifting my pen. Of course, it could do
some weird stuff. OK, let me just draw here. So here's a point, a, and here
is my function, f, and there is f of a. So I'm saying that the function
could come along, it could come down pretty steeply,
but it will get to that point. It might go on, steeper below,
or it might turn back. Or it might be level. But I can draw the whole
thing continuously. But now that description
with a piece of chalk isn't quite enough. And there's a formal definition that I have to explain. And it involves this same
idea of epsilon, this same idea of a strip. It means that if I take a little
strip around f of a-- so here's f of a plus a little
bit, and here's f of a minus a little bit-- then that's continuous. That function is continuous,
because-- now, remember, epsilon could
be smaller than I drew it, smaller than I can draw it,
but still positive. Then the requirement is that it
has to get near a, it has to get inside that band
and stay there. It can bounce all
over the place. But near the point, it's
got to get close. And now, how do I express that
in terms of epsilon? OK, well, there's a famous
description. Yeah, what do I mean by get in
there and stay in there? Ah! Can I just make a story? I'm going to use two Greek
letters, epsilon and delta, hated by all calculus students
and professors too, if they're truthful. OK, so the story goes,
we choose a band. Ah, since their Greek letters,
Socrates chooses epsilon. OK. So he's going to make it hard. He's going to make a
narrow band there. And then the function has got to
get into that band and stay there, close to a. OK, so what do I mean
by close to a? Well, that's where
delta comes in. That's, let's say, Socrates's
student, Plato. Then Plato can pick his number,
delta, which will be the width-- see, he says, OK, if you get
really close, I've got you. So he's trying to
please Socrates. So he says, woo, sorry, a had
better be in there somewhere. Now these bands are getting so
close, my a is, of course-- this is really a plus delta, and
this guy is a minus delta. Are you kind of with me? The logic goes, for any epsilon
chosen by Socrates, Plato can find a positive
delta-- epsilon, of course, was some
positive number, delta might be an extremely small
positive number-- so that if the distance
to a is smaller than Plato's distance-- so if we're in that
vertical band-- then we're in Socrates's
horizontal band. Then this f of x minus f
of a is below epsilon. So Socrates sets up any tough
requirement, any horizontal band, and then Plato meets
that requirement, if the function is continuous, by
choosing a vertical band that keeps everything inside
Socrates's band. Do you see that? Well, it takes some thought. It takes some practice, and as
always, it's not usually very hard to tell if a function
is continuous. Let me show you one
that isn't. A famous function that
is not continuous. Here's the sine of 1 over
x as x going to 0. What happens to the sine of
1 over x when x goes to 0? Well, the sine, we know,
oscillates minus 1, plus 1, minus 1, plus 1. But when it's a sine of 1 over
x, that oscillation really takes off, because if x gets
small, 1 over x is quickly getting larger. You're running along the sine
curve in a faster and faster and faster way. I can't draw it. Here's 0. But it's not staying
inside a band. Even with epsilon equalling 1/2,
Socrates has got Plato. Plato can't keep it in a band of
1/2 up and 1/2 down because the sine doesn't stay there. So there's a function that's
not continuous. I could make it continuous by
changing the function a little, maybe x times
sine of 1 over x. That would bring the
oscillations down and work. So there you go. That's epsilon and delta. And it takes a little
practice. And I just have to remember-- when you feel that the whole
thing is a bad experience-- some pity for a Socrates, who
actually took poison. Not because Plato gave him one
that he couldn't do, for some completely different reason. But this is the meaning of a
continuous function, and by getting that meaning which took
hundreds of years to see. And it takes some time to get
these two different things, to get the logic straight. If x is close to a, then f
of x is close to f of a. That's what this means, f
of x approaching f of a. That's what Socrates and Plato
together had to explain. OK, thank you. ANNOUNCER: This has been
a production of MIT OpenCourseWare and
Gilbert Strang. Funding for this video was
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