Epsilon-delta limit definition 1 | Limits | Differential Calculus | Khan Academy
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Epsilon-delta limit definition 1 | Limits | Differential Calculus | Khan Academy


Let me draw a function
that would be interesting to take a limit of. And I’ll just draw it visually
for now, and we’ll do some specific examples
a little later. So that’s my y-axis,
and that’s my x-axis. And let;s say the function
looks something like– I’ll make it a fairly
straightforward function –let’s say it’s a line,
for the most part. Let’s say it looks just
like, accept it has a hole at some point. x is equal to a, so
it’s undefined there. Let me black that point
out so you can see that it’s not defined there. And that point there
is x is equal to a. This is the x-axis, this is
the y is equal f of x-axis. Let’s just say
that’s the y-axis. And let’s say that this
is f of x, or this is y is equal to f of x. Now we’ve done a bunch
of videos on limits. I think you have an
intuition on this. If I were to say what is the
limit as x approaches a, and let’s say that this
point right here is l. We know from our previous
videos that– well first of all I could write it down –the
limit as x approaches a of f of x. What this means intuitively is
as we approach a from either side, as we approach it from
that side, what does f of x approach? So when x is here,
f of x is here. When x is here, f
of x is there. And we see that it’s
approaching this l right there. And when we approach a from
that side– and we’ve done limits where you approach from
only the left or right side, but to actually have a limit it
has to approach the same thing from the positive direction and
the negative direction –but as you go from there, if you pick
this x, then this is f of x. f of x is right there. If x gets here then it goes
here, and as we get closer and closer to a, f of x approaches
this point l, or this value l. So we say that the limit
of f of x ax x approaches a is equal to l. I think we have that intuition. But this was not very, it’s
actually not rigorous at all in terms of being specific
in terms of what we mean is a limit. All I said so far is as
we get closer, what does f of x get closer to? So in this video I’ll attempt
to explain to you a definition of a limit that has a little
bit more, or actually a lot more, mathematical rigor than
just saying you know, as x gets closer to this value, what
does f of x get closer to? And the way I think about it’s:
kind of like a little game. The definition is, this
statement right here means that I can always give you a range
about this point– and when I talk about range I’m not
talking about it in the whole domain range aspect, I’m just
talking about a range like you know, I can give you a distance
from a as long as I’m no further than that, I can
guarantee you that f of x is go it not going to be any further
than a given distance from l –and the way I think about it
is, it could be viewed as a little game. Let’s say you say, OK Sal,
I don’t believe you. I want to see you know, whether
f of x can get within 0.5 of l. So let’s say you give me 0.5
and you say Sal, by this definition you should always
be able to give me a range around a that will get f of
x within 0.5 of l, right? So the values of f of x are
always going to be right in this range, right there. And as long as I’m in that
range around a, as long as I’m the range around you give me, f
of x will always be at least that close to our limit point. Let me draw it a little bit
bigger, just because I think I’m just overriding the same
diagram over and over again. So let’s say that this is f of
x, this is the hole point. There doesn’t have to be a hole
there; the limit could equal actually a value of the
function, but the limit is more interesting when the function
isn’t defined there but the limit is. So this point right here– that
is, let me draw the axes again. So that’s x-axis, y-axis x,
y, this is the limit point l, this is the point a. So the definition of the limit,
and I’ll go back to this in second because now that it’s
bigger I want explain it again. It says this means– and this
is the epsilon delta definition of limits, and we’ll touch on
epsilon and delta in a second, is I can guarantee you that
f of x, you give me any distance from l you want. And actually let’s
call that epsilon. And let’s just hit on
the definition right from the get go. So you say I want to be no more
than epsilon away from l. And epsilon can just be any
number greater, any real number, greater than 0. So that would be, this distance
right here is epsilon. This distance there is epsilon. And for any epsilon you give
me, any real number– so this is, this would be l plus
epsilon right here, this would be l minus epsilon right here
–the epsilon delta definition of this says that no matter
what epsilon one you give me, I can always specify a
distance around a. And I’ll call that delta. I can always specify
a distance around a. So let’s say this is delta
less than a, and this is delta more than a. This is the letter delta. Where as long as you pick an x
that’s within a plus delta and a minus delta, as long as the x
is within here, I can guarantee you that the f of x, the
corresponding f of x is going to be within your range. And if you think about it
this makes sense right? It’s essentially saying, I can
get you as close as you want to this limit point just by– and
when I say as close as you want, you define what you want
by giving me an epsilon; on it’s a little bit of a game
–and I can get you as close as you want to that limit point by
giving you a range around the point that x is approaching. And as long as you pick an x
value that’s within this range around a, long as you pick an x
value around there, I can guarantee you that f of x will
be within the range you specify. Just make this a little bit
more concrete, let’s say you say, I want f of x to be within
0.5– let’s just you know, make everything concrete numbers. Let’s say this is the number 2
and let’s say this is number 1. So we’re saying that the limit
as x approaches 1 of f of x– I haven’t defined f of x, but it
looks like a line with the hole right there, is equal to 2. This means that you can
give me any number. Let’s say you want to try it
out for a couple of examples. Let’s say you say I want f of x
to be within point– let me do a different color –I want f
of x to be within 0.5 of 2. I want f of x to be
between 2.5 and 1.5. Then I could say, OK, as long
as you pick an x within– I don’t know, it could be
arbitrarily close but as long as you pick an x that’s –let’s
say it works for this function that’s between, I don’t
know, 0.9 and 1.1. So in this case the delta from
our limit point is only 0.1. As long as you pick an x that’s
within 0.1 of this point, or 1, I can guarantee you that your
f of x is going to lie in that range. So hopefully you get a little
bit of a sense of that. Let me define that with the
actual epsilon delta, and this is what you’ll actually see in
your mat textbook, and then we’ll do a couple of examples. And just to be clear, that
was just a specific example. You gave me one epsilon and I
gave you a delta that worked. But by definition if this is
true, or if someone writes this, they’re saying it doesn’t
just work for one specific instance, it works for
any number you give me. You can say I want to be within
one millionth of, you know, or ten to the negative hundredth
power of 2, you know, super close to 2, and I can always
give you a range around this point where as long as you pick
an x in that range, f of x will always be within this range
that you specify, within that were you know, one trillionth
of a unit away from the limit point. And of course, the one thing
I can’t guarantee is what happens when x is equal to a. I’m just saying as long as you
pick an x that’s within my range but not on a, it’ll work. Your f of x will show up to be
within the range you specify. And just to make the math
clear– because I’ve been speaking only in words so far
–and this is what we see the textbook: it says look, you
give me any epsilon greater than 0. Anyway, this is a
definition, right? If someone writes this they
mean that you can give them any epsilon greater than 0, and
then they’ll give you a delta– remember your epsilon is how
close you want f of x to be to your limit point, right? It’s a range around f of x
–they’ll give you a delta which is a range
around a, right? Let me write this. So limit as approaches a
of f of x is equal to l. So they’ll give you a delta
where as long as x is no more than delta– So the distance
between x and a, so if we pick an x here– let me do another
color –if we pick an x here, the distance between that value
and a, as long as one, that’s greater than 0 so that x
doesn’t show up on top of a, because its function might be
undefined at that point. But as long as the distance
between x and a is greater than 0 and less than this x
range that they gave you, it’s less than delta. So as long as you take an x,
you know if I were to zoom the x-axis right here– this is a
and so this distance right here would be delta, and this
distance right here would be delta –as long as you pick an
x value that falls here– so as long as you pick that x value
or this x value or this x value –as long as you pick one of
those x values, I can guarantee you that the distance between
your function and the limit point, so the distance between
you know, when you take one of these x values and you evaluate
f of x at that point, that the distance between that f of x
and the limit point is going to be less than the
number you gave them. And if you think of, it seems
very complicated, and I have mixed feelings about where
this is included in most calculus curriculums. It’s included in like the, you
know, the third week before you even learn derivatives, and
it’s kind of this very mathy and rigorous thing to think
about, and you know, it tends to derail a lot of students and
a lot of people I don’t think get a lot of the intuition
behind it, but it is mathematically rigorous. And I think it is very valuable
once you study you know, more advanced calculus or
become a math major. But with that said, this
does make a lot of sense intuitively, right? Because before we were talking
about, look you know, I can get you as close as x approaches
this value f of x is going to approach this value. And the way we mathematically
define it is, you say Sal, I want to be super close. I want the distance to be
f of x [UNINTELLIGIBLE]. And I want it to be
0.000000001, then I can always give you a distance around x
where this will be true. And I’m all out of
time in this video. In the next video I’ll do some
examples where I prove the limits, where I prove some
limit statements using this definition. And hopefully you know, when we
use some tangible numbers, this definition will make a
little bit more sense. See you in the next video.

About James Carlton

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100 thoughts on “Epsilon-delta limit definition 1 | Limits | Differential Calculus | Khan Academy

  1. This would have been so beneficial if Kahn would have used a marker board or a chalk board. This computer program is really sloppy and messy. I would love to see all these video lessons where he writes real neat on a marker board. He is very intelligent and a great teacher but the writing is far too messy and sometimes it cannot be interpreted. 

  2. I've watched a bunch of videos about the topic and it's crystal clear to me that if the limit of f(x) is within epsilon of L, then x is within delta of a. I can visualize it, express it mathematically and it makes perfect sense, however, how does that prove a limit???

  3. I got the concept.I got everything xplained n d video. But can any1 plz tel me whts the use of epsilondelta where is it applied for real?

  4. This is kind of obvious and simple, which makes me think that maybe I'm not understanding it fully? Basically it means that given an epsilon greater than zero, you can always find a delta greater than zero which will evaluate f(x) between that given epsilon and the limit, right? That simple?

  5. I have to seriously question the value of these so called "precise definitions of the limit", especially in the context of a first year differential calculus course. In fact, the very concepts associated with rigorous mathematics are quite questionable in consideration of Godel's Theorem.

    Here is all you need to know about the meaning of a limit. When we say that we are taking the limit of a function as x approaches c, we simply mean that we are letting x get extremely close to c, without x actually being equal to c. In short, if x were to be equal to c then our concept becomes inconsistent.

    Consider the definition of the derivative: the limit as delta x approaches 0 of the change in y over the change in x equals dy/dx. If delta x were to actually be equal to zero, we would be dividing the average value of y by zero, and the whole damn thing would blow up. So we say that delta x approaches 0, which means we're getting as close as we possibly can to zero, without actually getting to that last notch on the real number line that is zero. In essence, this would always make delta x some nonzero real number. The significance of those ideas, is that we are always finding an approximate solution (although as delta x gets closer to zero, we are getting much, much closer to the actual value of the limit.) Because we know that delta x can't be zero, when we work through our definition of the derivative and eliminate the delta x terms by plugging zero in for them, we are making an inconsistent statement. Plugging zero in makes our math easier, but we must understand that by doing so, we are still only approximating, and we are violating the limiting concept.

    The very notion that there is a "precise definition of the limit", that would allow us to find an exact value for limiting cases is inconsistent. Do not worry so much about these so called "precise definitions" and rigorous mathematics. Those ideas certainly didn't bother Newton or Leibniz, and all they did was invent calculus. And remember, rigorous work leads to rigor mortis. 🙂

  6. This tripped me up and I'm in multivariable calculus right now. I'm a couple sections behind because I spent too much time on this part with two variables. I really wish this would have come later.

  7. what is the purpose of this? why would you want to find an interval when you can just find the limit of the function and have an exact value instead of a set of numbers where one of them is correct?

  8. listening to this guy is quite cumbersome, since he is barely able to finish a sentence. quit starting sentences over , it will improve comprehendability. just a hint

  9. Hey do you think you could update this video? I find it less intuitive than the other videos. I understand it, but this video needs updated compared to the others. However, I am eternally thankful for K.A.

  10. so essentially calculus is simple algebriac inequalities, functions, factoring, and arithmatic with greek letters.

  11. Wasting to much time saying the same thing over and over again. Sal, you need to keep your videos more precise. I am not expecting the Feynman Lectures but we come to watch your videos for fun, not for wasting time.

  12. Pretty bad video, off to find another that doesn't have someone repeat themselves 20 times will making no progress.

  13. Thank you a lot!!!! I didn't get this concept at all at first, but now I actually get it! Thank you so much for the videoo

  14. Dude you need to redo this this topic or video, one the quality sucks ass. And secondly, your method of conveying the information is just all over the place. Speak clearly, simply, and precisely. Stop confusing yourself. It’s simple, explain the topic and that’s it. You keep conversing broken ass sentences and it’s hella distracting. Otherwise, I feel you’d really be able to bring the point home as you’re unequally brilliant and very selfless!

  15. The quality actually loses my attention man, your videos are helpful but occasionally i run into one i really need and it looks like this :/ idk if its an update that it needs, but Sal im lost 🙁

  16. I understand that we start with an epsilon…but why do we start with an epsilon? That never quite made sense to me…

  17. I understood the idea behind, but I couldn't understand how sal derived that linear inequality equation…
    To understand that equation I have even gone through the linear inequality playlist but still its vague for me😞😞😞😞😞

  18. Helped a lot for my college. This thing hardly went through my brain while in class, lmao. Especially its problems lol

  19. Don't you need to specify that if the delta interval gets smaller, the epsilon interval must get smaller???

    What if delta can get really small but epsilon has to stay big. Then this definition say that it's still a limit, when the function might not be approaching anything.

  20. thanks for the video, learning this on my own, i think you helped me somewhat acquire the understanding of this subject.

  21. Sal Khan – One of best people on the internets, period. He explains it far, far better than my $120,000 professor ever could.

  22. If you pick a range that limx->a f(x) falls into on the y axis greater than 0, there is a corresponding positive range about a on the x axis, if the limit exists.

    If the limit does not exist, you will never find a corresponding range about a on the x axis no matter how small you pick the range about f(x) on the y axis.

    Is this where convergent/divergent limits come from? As Epsilon gets smaller, delta gets smaller. If delta gets bigger as e gets smaller, the limit diverges: the limit does not exist.

  23. My math book stated that the distance between f(x) and L are less than the epsilon divided by 2? Can anyone help me on that plsss

  24. thank god for resources like this. i remember when people that were good at math kept all the information secret.

  25. when you're taking calculus over the summer, and he says they teach you this over the first three weeks and my professor gave it to me the first day 🙂

  26. At 4:55 Sal is pretty much like, “We’ll talk about epsilon and delta later, nvm we’re taking about them now” 😂😂

  27. Here’s one thing I don’t understand: why is are the distances before and after the limit L equal (epsilon), while the distances before and after the c value (delta) are also equal? This seems to fit more accurately a line than a curve where the rate of change changes. If you pick 2 f(x) values on a curve of equal distance from L and then find their corresponding x values, those x values will not be equidistant from the c value that corresponds to the limit L.
    I’m having trouble getting past this.

  28. If we pick any value which is less than the delta (means within the range of the delta value) then the distance of f(x) and the limit is less than the epsilon. Does this means that if we substitute the delta value, it will have an f(x)=Epsilon?

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