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Current time:0:00Total duration:6:59

Formal definition of limits Part 3: the definition

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 to 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 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 you you tell me how close you want 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 it you want f of X to be to L so you give a positive 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 okay you've given me that epsilon I'm going to find you I will find you another number find another positive number another number which we'll call Delta the lower case Delta the Greek letter Delta such that so I'll say where if X is within 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 see this second definition which I kind of made is 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 is then to find a delta where as long as X is within Delta of C that f of X will be within epsilon of the limit so that is doing it it's saying look if we were constraining X in such a way that if X is in that range to see that 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 so this right over here this right over here would this point right over here is our limit plus epsilon and this right over here might be our limit minus this right over here is limit minus Epsilon and you say okay sure I think I can do I can I can get your f of X within within this range of our limit and I can do that by defining a range around C and it really it could you know I could visually look at this boundary but I could even go narrow or that boundary I could I could go I can go right over here says okay 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 right this down is C minus Delta so I'll find you some Delta so that if you take any X you take any X in the range C minus Delta to C plus Delta and maybe the function is not even defined at C so we 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 meet your 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 a little bit more math notation so I'll write the same exact statement a little bit more a little bit more a little math you what's the exact same thing so you give me you give or let me write it this way given given given and epsilon greater than zero we can find so that's kind of the first part of the game we can find a delta greater than 0 such that such that if X if X is within Delta of C so what's another way of saying that X is 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 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 in this range between C minus Delta and C plus Delta and that's these are the X's that satisfy that right over here then then and I'll do this in a new color then the distance between your f of X 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 if if the limit truly does exist it truly is L is if you give me any positive number epsilon it could be a super super small one we can find a delta so we can define a range around C so that if we take any X 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 C so that's these points right over here that f of those X's that f of at the 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 is going to sit someplace 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
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