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Course: AP®︎/College Calculus AB>Unit 1

Lesson 17: Optional videos

Formal definition of limits Part 3: the definition

Explore the epsilon-delta definition of limits, which states that the limit of f(x) at x=c equals L if, for any ε>0, there's a δ>0 ensuring that when the distance between x and c is less than δ, the distance between f(x) and L is less than ε. This concept captures the idea of getting arbitrarily close to L. Created by Sal Khan.

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• why is epsilon always greater than delta?
(31 votes)
• It's not! Sometimes epsilon is greater than delta, sometimes delta is greater than epsilon.
(44 votes)
• Why are epsilon and delta must be always GREATER than 0? Why can't both of it be always GREATER or EQUAL to 0?
(17 votes)
• 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.
(39 votes)
• Is it true if I say that f(c+δ) = L+ε and f(c-δ) = L - ε , for any function ?
(13 votes)
• 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.
(20 votes)
• Why do we use epsilon and delta and not some other greek letter? Is there a specific reason for using these letters?
(9 votes)
• The choice of these letters is arbitrary, but there's a long tradition of using these letters for this purpose. In particular, epsilon is typically used whenever referring to an arbitrarily small amount. The famous mathematician Paul Erdos extended the concept, humorously referring to small children as epsilons.
(24 votes)
• 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)
• He uses the < sign because he isn't trying to find the limit, he is just trying to get close to the limit.
(5 votes)
• 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" ?
(4 votes)
• 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.
(7 votes)
• Can't there be multiple deltas for an epsilon?
(3 votes)
• 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)
• 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)
• 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)
• i understand why we need limits but what is the point of the epsilon delta definition
(1 vote)
• Newton could not rigorously prove any of his Calculus despite a lifetime of effort. The way continuous functions are introduced to beginning calculus students is to ask something intuitive like: "As x gets this close to c, how close does f(x) get to f(c)?” This hand-wavy approach gives students an intuitive feel for the topic, but is not useful for rigorous proofs. Many decades after Newton's struggles, the Bohemian mathematician Bolzano turned the question around by asking essentially: "If we want f(x) to get this close to f(c), how close does x have get to c?" This is the basis of the epsilon-delta definition, which finally allowed mathematicians to rigorously prove the calculus invented by Newton (and Leibnitz). It is counter-intuitive to math students at first, but after a while they internalize it and it becomes second nature. Remarkably, over such a long time the smartest mathematicians (including Newton) missed discovering that crucial one-liner definition. Analysis (of which calculus is a part) involves a lot of proving that something is within epsilon of something else. As a footnote, during my school days, another student remarked to me: "Bruce came within epsilon of flunking out last year".
(5 votes)
• Cant get this, if x is within delta of c, f(x) will be within epsilon of L. What does it mean?
(3 votes)
• It simply means that, If x lies between the range of delta (i.e. c and c + delta AND c and c - delta)
then f(x) (i.e. the value of y) will lie within the range of L and epsilon(i.e. L and L + epsilon AND L and L - epsilon)
(1 vote)

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.