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Current time:0:00Total duration:6:06

Formal definition of limits Part 2: building the idea

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 you can get f of X as close as close to L as you want as you want I'll put that in quotes right over here because it's kind of a little loosey goosey is how close is that but as close as you want by getting getting X sufficiently close sufficient ly close 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 see that if X isn't within that range that 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'll you draw another diagram so let's say that let's say that is my drawing 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 this is C and let's say 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 say I want it to be defined at C at least just for the it could be you can always find a limit even where it is defined but let's say our function looks something like that and it can even 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 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 is a hole it is undefined when X is equal to C and it even has a little kink in it just like that and 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 what's saying that you can get as can get f of X as close to L as you want as you want so if you tell someone I want to get within a certain 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 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 if 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 that you're claiming that 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'm going to I want to get within I want to get within 0.5 I want to get f of X within 0.5 of L so this right over here this right over here would be L plus 0.5 and this right over here is L L minus 0.5 and then you say fine I'm going to give you a range around see 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 see minus 0.25 and let's say that this value right over here is C plus 0.25 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 the corresponding f of X is going to sit in the range that you care about and you're saying okay that fine the you won that round let me make it even tighter maybe with instead of saying within 0.5 I want to get within zero 0.05 and then you'd have to do this exercise again and find another range and in order for this to be true in order for this to be true you have to be able to do this for any range 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 see 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 0.5 of L and you say well as long as X is within point two-five 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
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