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AP®︎/College Calculus AB
Course: AP®︎/College Calculus AB > Unit 1
Lesson 17: Optional videosFormal definition of limits Part 2: building the idea
Explore the rigorous mathematical definition of a limit as x approaches c, and understand how to get f(x) as close to L as desired by finding a range around c. Dive into the epsilon-delta definition and its application in proving limits for various functions. Created by Sal Khan.
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What does y=f(x) mean? I thought they were the same thing. Sorry if this is a stupid question.(100 votes)
John, there are no stupid questions. And actually, your question is a very good one.
In a strict mathematic sense, y is just a variable. When someone writes "y=f(x)", it means that the value of y depends on the value of x, which is another variable. That is, for different values of x, there is a function, called f(x), which determines the value of y.
The x variable is therefore called the "independent" variable, while the y variable is called the "dependent" variable because it's value "depends" on the value of x.
So, in short. The expression y=f(x) is basically a formal way of stating which one is the independent variable (in this case, x) and which one is the dependent one.(297 votes)
- why is it that the range of the L is 0.5 and the range around c was 0.25, my point is that will the range for the x-axis always be less than that for the range for the y-axis?(39 votes)
I'm pretty sure that it depends on the type of F(x) (the function) that you encounter. You can try with basic function at home, draw one and then find a casual range in Y and watch how it work on the X axis; you'll see that depends on the trend of the function.(36 votes)
At4:30, how did he find that the corresponding range for c is more or less 0.25 when the range for L is more or less 0.5?(16 votes)
He never defined the function; it's a hypothetical situation, and he randomly chose the interval (aka he could have chosen c + or - any constant here).(38 votes)
Why were epsilon delta forms created?(12 votes)- Initially, calculus was based on somewhat vague ideas about infinitesimally small quantities, and some scholars argued that these methods weren't valid. The epsilon-delta concept was created to provide a rigorous logical foundation for the methods used in calculus.(33 votes)
- Are limits used only in situations when x approaches the value for which the function f(x) is undefined, and not for other values?(7 votes)
Usually, but not necessarily.
It's completely fine to say lim{x->1}x=1, this is proper and completely allowed, but somewhat pointless.(16 votes)
- why should one be able to find range of c to prove that limit exist..?... what is basic idea .?(7 votes)
- it is a set up for the proof in the next video, and gets you thinking.(7 votes)
In what kind of circumstances would people use this Epsilon-Delta definition out of the studies of mathematics?(5 votes)- in determining what delta you should choose to make the certain error lim x>c f(c). example: how much delta from c-delta<x<c+delta should i choose to make my square area error 0.5.? (If we want f(x) to get this close to f(c), how close does x have get to c?)(10 votes)
- Around5:00you start talking about the range. What if the graph dramatically spikes within that range? Your graph is between L-0.5 and L+0.5 at all times but what if at L-0.25 it all of the sudden dramatically spikes out of L-0.5 just for a quick dip and comes back? Then it isn't within that range, then you can find an X within that range that doesn't give you a y within that range?
Sorry for the poorly phrased question.. I hope someone understood what I was trying to convey.(5 votes)- Interesting question. The way I was able to explain it to myself to understand was to change the terminologies all together. I used the concept of domain and ranges rather than using a bounded range around L and C. If you visualize the concept of the range of the function given the domain 0.25 < c < -0.25, you would be able to see that no matter how many spikes occur in that DOMAIN range, the value at that spike will INDEED always map to (or be mapped from?) a value within the original domain which was the boundary around c. You have to, in a way, step back from a calculus way of reasoning a little bit to a more basic algebraic way of reasoning. As long as the value y = f(x) exists (in this case, the value of the spike) within a domain (in this case + or - 0.25 off c), any value within the domain ( boundary around c) will always map to a value within its EQUIVALENT range (which in this case is the boundary around L). Since your spike has been introduced to the function, the range of the function can no longer be + or - 0.5 off L (even though the domain remained the same). The range has to be increased for the spike to be accounted for or else, the guy who is playing the game is trying to cheat by bending the rules of algebra a little bit. :) . Think of it this way, by introducing the spike, that value will cease to exist because it indeed doesn't fall within the range + or - 0.5 around L, therefore we will have to go back to the drawing board and redefine a value that would fall within the range specified.That's the way I was able to understand it. I hope it helped you because it actually made me understand algebra a little bit better which i thought was pretty neat. Math works that way! :)(4 votes)
- If it doesn't lie within the "range", does that mean the limit doesn't exist? Is it like saying if the limit doesn't approach the same point it doesn't exist? Thanks. You guys are awesome.(5 votes)
- yes. If there is no range for c where it gets closer to L within that range, the limit does not exist.(4 votes)
Are we just approximating the limit then since we're only getting "as close as we want"? I don't see how this is rigorous at all, it's pretty much the same thing as estimating limits from tables.
What I'm saying is, there doesn't seem to actually be a way to guarantee that the limit of some function f(x) is L. Even if you get infinitely close to L, you're still technically infinitely far away from it because infinity is infinite. Our little brains cant get into that sort of business.(3 votes)
'Infinitely close to L' isn't a thing in the real numbers. If we have two numbers, a and b, then the distance between them is either 0 (if a=b) or some finite, positive number |a-b|. There are still infinitely many numbers between a and b, but that doesn't mean the distance between them is infinite.
Because of this, we know limits are uniquely defined. If there were two viable limits, L, M, then they are some distance |L-M| apart. If L and M were limits, the function would get 'as close as we like' to both of them. Specifically, the function values would get within |L-M|/2 of both L and M, which is impossible.
So there cannot be multiple values of a limit, even very close to each other. The limit of f(x) is a uniquely specified real number.(3 votes)
4:30Video transcript
Let's try to come up with
a mathematically rigorous definition for what
this statement means. The statement that
the limit of f of x as x approaches c
is equal to L. So let's say that this means that
you can get f of x as close to L as you want. I'll put that in
quotes right over here, because it's kind
of a little loosey goosey as how close is that. But as close as
you want by getting x sufficiently close to c. So another way of saying
this is, if you tell me, hey, I want to get my f of x to
be within 0.5 of this limit. Then you're telling me if
this limit is actually true, you should be able to
hand me a value around c. That if x is within
that range, then f of x is definitely going to be
as close to L as I desire. So let me draw that out to
make it a little bit clearer. And I'm going to zoom in. I'm going to draw
another diagram. So let's say that this right
over here is my y-axis. And I'm going to zoom in. I'm going to draw a slightly
different function, just so we can really focus on what's
going on around here. The range is around c, and the
range is around L. So that's x. This right over here is y. Let's say that this is c. And let's just zoom
in on our function. So let's say our function
looks, is doing something like, let's say it does something
like, let's see, I don't want it to
be defined at c. At least just for
the-- it could be. You can always find a limit
even where is defined. But let's say our function
looks something like that. And it can have a little kink
in it, the way I drew it. So it looks something like this. It's undefined. Let me draw it a
little bit different. So it is undefined
when x is equal to c. So this is the point
where there's a hole. It is undefined when
x is equal to c. So it even has a little
kink in it, just like that. And what we want to do
is prove that the limit, as x, the limit of f of x--
and let me make it clear, this is the graph
of y is equal to f of x-- we want to get an idea
for what this definition is saying. If we're claiming that the limit
of f of x, as x approaches c, is L. So conceptually, we
get the gist already. We already get the gist that
this right over here is L. But what is this
definition saying? Well, it's saying that you
can get f of x as close to L as you want. So if you tell
someone, I want to get f of x within a
certain range of L, then if this limit
is actually true, if the limit of f of
x as x approaches c really is equal to
L, then they should be able to find
a range around c. That as long as x is
around that range, your f of x is going to be
in the range that you want. So let me actually go
through that exercise. It really is a little
bit like a game. So someone comes up to
you and says, well, OK. I don't necessarily
believe that you're claiming the limit of f of x as
x approaches c is equal to L. I'm not really sure
if that's the case. But I agree with
this definition. So I want to get within 0.5. I want to get f of x within 0.5
of L. So this right over here would be L plus 0.5. And this right over
here is L minus 0.5. And then you say, fine. I'm going to give
you a range around c, that if you take any x within
that range, your f of x is always going to fall in
this range that you care about. And so you look at this-- and
obviously we haven't explicitly defined this function. But you can even eyeball it, the
way this function is defined. It won't be that easy
for all functions. But you look at it like this. And you say that this
value, just the way it's drawn right
over here, let's say that this is c minus 0.25. And let's say that this value
right over here is c plus 0.25. And so you tell
them, look, as long as you get x within
0.25 of c, so as long as your x's are sitting
someplace over here, the corresponding f of x is
going to sit in the range that you care about. And you say, OK, fine. You won that round. Let me make it even tighter. Maybe instead of saying within
the 0.5, I want to get within is 0.05. And then you'd have to do
this exercise again and find another range. And in order for
this to be true, you would have to be able
to do this for any range that they give you. For any range around
L that they give you, you have to be able to get
f of x within that range by finding a range around c. That as long as x is
that range around c, f of x is going to
sit within that range. So I'll let you think
about that a little bit. There's a lot to think about. But hopefully this made sense. We did it for the particular
example of someone hands you the 0.5, I want
f of x within the 0.5 of L, and you say, well, as long
as x is within 0.25 of c, you're going to match it. You need to be able to
do that for any range they give you around
L. And then this limit will definitely be true. So in the next video, we
will now generalize that. And that will really bring us
to the famous epsilon delta definition of limits.