Stony Brook MAT 125 Spring 2015
Definition of the Limit (optional material)

Start   now i want to give you a formal definition of the limit this is not strictly required for what you need to do in this course but as mathematicians we need to insist upon defining everything carefully and the definition that we've given which is essentially we say that the limit as X goes to some number a is l means in a very common kind of squishy sense that i
0:37can make F of X as close to l as I want by making x close enough to a so that's the
1:06definition that we are using in this class f of x gets close to el if we make X close enough to a but that's not a very formal precise definition so in terms of the picture this is saying here at some point a if this is the graph at this is the x-axis your f of X we have our graph coming in here we don't care what happens to a we
1:37don't get to know that this value L is here and so let's say well whatever f does if I look close enough here then it's really close so if i look just really close here i can make it really close here to formalize this and make it
2:00more official we just sort of say what we mean by i can make it as close as I want in a more careful way so what is close mean means means that we want to make F of X minus L to be as close to zero but not necessarily equal to 0 but as close to zero as we want and
2:32in order to make that happen we choose x close enough to a so going to erase this now so what that means you tell me a number of Tolerance i'm going to call h for height and then I can ensure i want an H needs to be positive but any h you tell me and then
3:12I want to find some w for with so that if X is well that's right this way first x is
3:35between a plus W and a minus w then f of X will be between l plus h minus H and if this works for every hu tell me then we say that the limit is l
4:05it has to work for every age which is not 0 and here also this is true Long ignore what happens in it we can rewrite this well maybe this is good enough let's get rid of the you tell me blah blah stuff so this means you tell me a tolerance H we really say for every h bigger than 0
4:44I want to find W so that means there is something so
5:05whenever and i'm going to write the distance from X to a is less than H and let's also so that I have to say if x is not equal to a and not zero so if hoops that's W if X minus a is less than w then FX minus L will be less
5:45than H so this is the formal definition the official mathematical definition of the limit you tell me how close you want it and I can make it work in terms of a picture that's saying blow it up a little bit my function does whatever it
6:07does here's my value a you want to get you give me this distance the distance above and below my desire type L and I want to trap the function the graph i want to trap the graph of the function inside a little box and no matter what with you get wet what height you give me here I can choose a wieth that traps the function in
6:35the box let me do an example of this just to show and I'll do a stupid example well it's not stupid but i'll do an example where we know it's already true let's take say x squared so we already know that the limit as X goes to 2 of x
7:02square is 4 like I said this is kind of silly because we know how squaring works we already know that we can get this limit just my plug in but let's turn it into the definition that I already erased so this means you tell me so i need to find a relationship between the height that you give me h and the width that I need to make it happen w so i need to find so given some
7:40h and we're thinking of h is like . 01 or something like that must fine w so that whenever X whenever X is within w to f of X is within h4 so this is pretty
8:17straight forward this is just saying we draw this picture here's the graph of y equals x squared here's two heres 4 or not exactly to scale the close enough for me and we're just
8:33saying you give me some distance here this is my h there's four and I want to find some width here that forces it to be in that little box well that's not too hard so what I need to do is I need to make sure that if i start here I wind up there so i want to say i want to be I want to be less than
9:074 plus or minus h and in order to be less than 4 plus or minus h I need to start your to going to add a little bit some with and square and I want to make sure that two plus W screen 2 plus W will be less square we less than 4 plus h and I want to solve it for w
9:31ok so square it out 4 plus 4w its really plus or minus but ok for plus or minus 4 w plus w square want that to be less than 4 plus h then I just bring in all over to one side that's what I want since this function is a pretty nice function there's no like big pokes up down or poke poke up or down it's a nice monotonic function I can just figure out when it's zero and that
10:03will tell me the things i got a 4 on both sides kill those off bring this H over here i get w square plus more maybe minus 4 w minus 4 no minus H no one is that zero ok so now this is a quadratic so i can use the quadratic formula that's negative B which is four plus or minus the square root of b squared
10:34that's 16 minus 4ac a is one C is negative H so that plus h over to those are the w's that do it that are zero in terms of H clean this up a little bit that's the same as negative 2 plus or minus the square root there's a 4 here that I lost that can pull this 4 out of each
11:00so that gives me 4 plus h so that's 4 plus h so what does that mean that means that to make this happen so that means must have I want to choose w so that w is less than square root for plus h minus 2 and that will do the trick for
11:36me I don't really care what this formula is they just care that there is a formula so i can do this and as long as I choose w and this way everything will be great and I'll see that the limit will actually be to this seems like probably two you like a bunch of formalistic nonsense but it really let's us say
12:00precisely what we mean by i can make this close to force this to be as close as I like and that's the formal definition again you won't be asked to do this in math 125 or bath 131 but this is really what's going on underneath the hood inside the carburetor so once again that the definition says the limit as X goes to a FX is l means for every positive h there is some w so that
12:44whenever X is not exactly at a but it's less than w
13:01we have f of x with in h of l that's the actual definition now usually one doesn't use h1 uses the Greek letter epsilon and one doesn't use w one uses
13:31the Greek letter Delta but it doesn't matter what letters we use i know that some people find everyone's and Delta's scary so i used hs snd w's but you know we could just as well say for every and top hat is supposed to be a half doesn't look much like a half does it for every top hat that we choose there is a smiley
14:05face doesn't matter what symbols we use you can use top hats and smiley faces you can use epsilon delta is you can it use agents and w's whatever works for you so the reason that i mentioned this stuff about epsilon Delta's is because this is sometimes call Epsilon Delta Delta Epsilon
14:35this is sometimes called the epsilon Delta definition of fun use whatever works for you in fact don't use anything because I don't require you to know this in class