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Formal definition of limits Part 2: building the idea

Some background intuition to make the formal definition of a limit make intuitive sense. Created by Sal Khan.

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  • starky ultimate style avatar for user Jade Bardai
    At , 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?
    (14 votes)
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  • orange juice squid orange style avatar for user jonah.yoshida
    Why were epsilon delta forms created?
    (11 votes)
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    • blobby green style avatar for user Creeksider
      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.
      (27 votes)
  • orange juice squid orange style avatar for user Richik Bandyopadhyay
    Are limits used only in situations when x approaches the value for which the function f(x) is undefined, and not for other values?
    (6 votes)
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  • old spice man green style avatar for user Suraj Jain
    why should one be able to find range of c to prove that limit exist..?... what is basic idea .?
    (6 votes)
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  • leaf blue style avatar for user Matthew Jeon
    In what kind of circumstances would people use this Epsilon-Delta definition out of the studies of mathematics?
    (4 votes)
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  • blobby green style avatar for user San Red
    Around you 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)
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    • blobby green style avatar for user John Uduma Junior
      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)
  • blobby green style avatar for user Antonio Maestas
    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)
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  • leafers ultimate style avatar for user Andrew Escobedo
    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)
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    • leaf green style avatar for user kubleeka
      '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)
  • blobby green style avatar for user Tom
    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?
    (2 votes)
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    • male robot hal style avatar for user bruno.garcia
      It depends on the function. Watch the video where Sal proves a limit using epsilon-delta definition. There, the function turns out to be f(x)=2x and the values of epsilon was 0.5 and delta 0.25, just like in this video here (he knew what was coming).
      (4 votes)
  • purple pi teal style avatar for user Affshafee
    What if my L (or limit) is a maximum point, let's say 4 in the y axis for the curve y= -x^2 + 4x +2? Then what value will be greater than L? I can get a value less than L. But what about greater than L?
    (3 votes)
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Video 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.