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Formal definition of limits Part 3: the definition

The epsilon-delta definition of limits says that the limit of f(x) at x=c is L if for any ε>0 there's a δ>0 such that if the distance of x from c is less than δ, then the distance of f(x) from L is less than ε. This is a formulation of the intuitive notion that we can get as close as we want to L. Created by Sal Khan.

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  • blobby green style avatar for user Tom
    why is epsilon always greater than delta?
    (1 vote)
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  • leaf grey style avatar for user Jakob Vtv
    Why are epsilon and delta must be always GREATER than 0? Why can't both of it be always GREATER or EQUAL to 0?
    (16 votes)
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    • blobby green style avatar for user Pete Halton
      Because limits have to do with numbers being within a certain (positive) distance of other numbers, and epsilon and delta are (positive) distances. If the distance between two numbers is zero, the numbers are equal, and so they are trivially within any positive distance of each other, but that is not what the idea of limits is about.
      (32 votes)
  • piceratops ultimate style avatar for user William A. Suzukayama
    Is it true if I say that f(c+δ) = L+ε and f(c-δ) = L - ε , for any function ?
    (12 votes)
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    • ohnoes default style avatar for user Tejas
      No it is not true. Assume we have a function f(x)=x^2. Obviously, the limit as x->0 is 0, so L=0. We can't have any negative y values, in this function, so there is no such thing as L-ε, because that would be negative.
      (19 votes)
  • duskpin sapling style avatar for user Rohini
    Why do we use epsilon and delta and not some other greek letter? Is there a specific reason for using these letters?
    (10 votes)
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  • duskpin ultimate style avatar for user Jinsung Lim
    How can we define one-sided limits?
    By using "x-c < -d and x < c" or "x-c < d and x > c" instead of "|x-c| < d" ?
    (5 votes)
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    • leaf grey style avatar for user Qeeko
      I will not formulate the most general way of defining one-sided limits (it requires some knowledge of point-set topology), but suppose ƒ is a real-valued function defined on a set containing an open interval of the form (a, b), where a < b are two real numbers.

      I) One says that ƒ has right-sided limit L at a if and only if there for every real number ε > 0 exists a real number δ > 0 such that |ƒ(x) - L| < ε for all real numbers x such that a < x < a + δ. (This latter condition on x may be rephrased as 0 < x - a < δ.) In other words, the condition |ƒ(x) - L| < ε is to hold for all x in the interval (a, a + δ) for some δ > 0.

      II) One says that ƒ has left-sided limit L at b if and only if there for every real number ε > 0 exists a real number δ > 0 such that |ƒ(x) - L| < ε for all real numbers x such that b - δ < x < b. (This latter condition on x may be rephrased as -δ < x - b < 0, i.e., 0 < b - x < δ.) In other words, the condition |ƒ(x) - L| < ε is to hold for all x in the interval (b - δ, b) for some δ > 0.

      There is a more general notion of one-sided limits. If ƒ is defined on a set X of real numbers, and if p is a limit point of the intersection of X with (p, +∞), we say that ƒ has right-sided limit L at p if and only if for all ε > 0 there exists δ > 0 such that |ƒ(x) - L| < ε for all x in X with p < x < p + δ. One defines left-sided limits similarly.
      (6 votes)
  • leaf red style avatar for user CalvinReedy
    Why does he use a < sign rather than a less than or equal to sign to say that x is within delta of c or to say that f(x) is within epsilon of L? Wouldn't the x value at the point (c+or-delta,f(c+or-delta)) be within the required range?
    (5 votes)
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  • leaf green style avatar for user Amita
    Can't there be multiple deltas for an epsilon?
    (3 votes)
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    • leaf grey style avatar for user Qeeko
      Let 0 < δ' < δ be real numbers, and let x and c be real numbers. Observe that |x - c| < δ' implies that |x - c| < δ. Hence 0 < |x - c| < δ' implies 0 < |x - c| < δ. Therefore, if you have found one δ for which the condition holds, then any other δ' less than δ also works.
      (8 votes)
  • blobby green style avatar for user Mujtaba Hussain
    The definition of limits provided assumes that f(x) is defined for all real numbers, but if f(x) is not defined for all real numbers, then ε cannot be any number you want which is greater than zero. If f(x) is not defined on f(c) itself, it's not a problem since x cannot equal c from definition 0 < |x - c| < δ anyway. But perhaps that isn't an issue in practice, since when f(x) is undefined, | f(x) - L | < ε is also undefined, and you can just ignore it? Any clarity as to how this definition can be used in practice would be appreciated. Am I right in thinking this definition does not apply when c=∞ or when L=∞?
    (2 votes)
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    • leaf green style avatar for user kubleeka
      The definition doesn't apply when c or L are infinite because we demand that f(x) have real inputs and outputs, and infinity isn't a real number.

      𝜀 can be chosen completely arbitrarily no matter where f(x) is defined. If 𝜀 is very large, larger than the range of f(x), then |f(x)-L| will just be much smaller than 𝜀. So what?
      (5 votes)
  • blobby green style avatar for user bubblegum001
    Why is the statement "if |x-c|<δ, then |f(x)-L|<ε" instead "if |x-c|≤δ, then |f(x)-L|≤ε?" It is a tiny question but I need to know why there is a less than sign and not a less than or equal to sign.
    (5 votes)
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    • blobby green style avatar for user petergarnaes
      Because it needs to be WITHIN the limit proposed. It is so by definition. The opponent picks an ε that he thinks is so close to L that if you can pick an x that means f(x) is even closer to L than ε (ie. closer than δ), then the opponent accepts he has been beaten.

      Accepting equality is of course not meaningful, that would mean our opponent would be satisfied even if we picked an x such that f(x) = ε he would consider himself beaten. picking an x such that f(x) = ε doesn't say anything meaningful, and is of course totally contradicting the method of proving the limit.
      (0 votes)
  • leaf grey style avatar for user Ibrahim Yousuf
    Cant get this, if x is within delta of c, f(x) will be within epsilon of L. What does it mean?
    (3 votes)
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Video transcript

In the last video, we tried to come up with a somewhat rigorous definition of what a limit is, where we say when you say that the limit of f of x as x approaches C is equal L, you're really saying-- and this is the somewhat rigorous definition-- that you can get f of x as close as you want to L by making x sufficiently close to C. So let's see if we can put a little bit of meat on it. So instead of saying as close as you want, let's call that some positive number epsilon. So I'm just going to use the Greek letter epsilon right over there. So it really turns into a game. So this is the game. You tell me how close you want f of x to be to L. And you do this by giving me a positive number that we call epsilon, which is really how close you want f of x to be to L. So you give a positive number epsilon. And epsilon is how close do you want to be? How close? So for example, if epsilon is 0.01, that says that you want f of x to be within 0.01 of epsilon. And so what I then do is I say well, OK. You've given me that epsilon. I'm going to find you another positive number which we'll call delta-- the lowercase delta, the Greek letter delta-- such that where if x is within delta of C, then f of x will be within epsilon of our limit. So let's see if these are really saying the same thing. In this yellow definition right over here, we said you can get f of x as close as you want to L by making x sufficiently close to C. This second definition, which I kind of made as a little bit more of a game, is doing the same thing. Someone is saying how close they want f of x to be to L and the burden is then to find a delta where as long as x is within delta of C, then f of x will be within epsilon of the limit. So that is doing it. It's saying look, if we are constraining x in such a way that if x is in that range to C, then f of x will be as close as you want. So let's make this a little bit clearer by diagramming right over here. You show up and you say well, I want f of x to be within epsilon of our limit. This point right over here is our limit plus epsilon. And this right over here might be our limit minus epsilon. And you say, OK, sure. I think I can get your f of x within this range of our limit. And I can do that by defining a range around C. And I could visually look at this boundary. But I could even go narrower than that boundary. I could go right over here. Says OK, I meet your challenge. I will find another number delta. So this right over here is C plus delta. This right over here is C minus-- let me write this down-- is C minus delta. So I'll find you some delta so that if you take any x in the range C minus delta to C plus delta-- and maybe the function's not even defined at C, so we think of ones that maybe aren't C, but are getting very close. If you find any x in that range, f of those x's are going to be as close as you want to your limit. They're going to be within the range L plus epsilon or L minus epsilon. So what's another way of saying this? Another way of saying this is you give me an epsilon, then I will find you a delta. So let me write this in a little bit more math notation. So I'll write the same exact statements with a little bit more math here. But it's the exact same thing. Let me write it this way. Given an epsilon greater than 0-- so that's kind of the first part of the game-- we can find a delta greater than 0, such that if x is within delta of C. So what's another way of saying that x is within delta of C? Well, one way you could say, well, what's the distance between x and C is going to be less than delta. This statement is true for any x that's within delta of C. The difference between the two is going to be less than delta. So that if you pick an x that is in this range between C minus delta and C plus delta, and these are the x's that satisfy that right over here, then-- and I'll do this in a new color-- then the distance between your f of x and your limit-- and this is just the distance between the f of x and the limit, it's going to be less than epsilon. So all this is saying is, if the limit truly does exist, it truly is L, is if you give me any positive number epsilon, it could be super, super small one, we can find a delta. So we can define a range around C so that if we take any x value that is within delta of C, that's all this statement is saying that the distance between x and C is less than delta. So it's within delta of C. So that's these points right over here. That f of those x's, the function evaluated at those x's is going to be within the range that you are specifying. It's going to be within epsilon of our limit. The f of x, the difference between f of x, and your limit will be less than epsilon. Your f of x is going to sit some place over there. So that's all the epsilon-delta definition is telling us. In the next video, we will prove that a limit exists by using this definition of limits.