well uh good afternoon again we're going to talk about section 13.2 right now you see we have this we had this concept uh it's how you start calc 1 it was called limits and you learn to loath them loath is the right word until you really learned them like okay this isn't this isn't so bad so here's what we're going to do we're going to talk about limits and continuity of multivariable functions we'll see how this works what I really want to focus on for about the first 20 minutes is what it means what are you doing when you're finding a limit because if you understand the concept how we're going to do the rest of it makes a lot more sense if you don't really get what we're doing like why are we doing this then they're not going to make sense at all you're just going to have a a rule of a set of rules that you're going to go through probably never really get it and then eventually make a whole lot of mistakes on it make sense so I want to focus on what a limit means when we're talking about multivariable functions let's start with something that we know let's start with something about limits of with functions in one variable and what we mean of course when I talk about variables I mean independent variables so when we have a function of one independent variable what we get is we get this curve on a plane that that's entirely what we have one variable two Dimensions remember talking about that from a couple times ago or last time one variable two Dimensions so if this is my function I say well how does a limit exist if we have a function in one independent variable we go well it exists if this limit equals a number okay and it doesn't exist if that equals like an ASM toote and they're going opposite ways or if we have different values but but more specifically we we say here's what has to happen and this is the big part okay because it's going to change when we get to multivariable functions here's how we recognize that limits exist for functions in one variable go well what well you know what you know it how do we tell that limits exist for functions of one variable what do we do how do you evaluate them what do you if I say does the limit exist here does it exist at the point P we yes or know yes why why does it exist does it exist at the value P here okay why because from from both sides both sides so we have this idea with limits of a left side and a right side you guys remember that it's a long time ago I know and that's how we determine that limits actually exist we say okay well if this limit exists what that implies or how we got that was that the limit as we approached along the axis of the independent variable from the left to a certain value and from the right to a certain value were the same so we do this hey let's come now here's here's the big thing here's a big thing this is what what what matters how where do we travel what do we travel along when we're checking the limit of a function of one variable what do we travel along well okay so we're moving this way but we travel along the actual curve we don't travel along the x-axis cuz if in both these situations let me make them a little bit more on top of each other okay so let's see here's the x-axis here's a there's two different functions here there's top function bottom function so hey uh what's the limit as we approach a you don't go like this you don't go um a done D for done deeper done that's what deeper done we don't go this way it's not what we do to check limits what do we travel along when we are evaluating our limits think back to what a limit means a limit means what's the height of your function what's the function value as we're approaching that number on that axis that's what it means what it implies is that we're approaching that number on that axis from both the left and the right and the height of our function has to be equal for that limit to exist you guys understand that concept I don't want to have to go back and read Cal one but I will pause right now we watch the videos all the This Together awesome I get to soothe my voice it hurts man it's hard work that's what it mean so do we actually travel along the xais No No we travel along the curve that's what we travel along so we go okay here here's how we do it from the left what's the height of my finger as we get close to our vertical dotted line which is that a okay it's right here what's the height of my finger is get close to that vertical dotted line ah it's at the same height that's how we know our limit exists what's the height of my finger oh it's it's over here as I'm traveling along the curve approaching this x value dotted line it's right here from the right hand side traveling along the curve it's right here wait a minute those things are different the height of my function the height of the curve is different as I'm approaching the same x value x-axis from left and right limit doesn't exist limit does exist show hands feel okay with that one okay now we extend the concept okay let's consider something else now this happens if this happens the limit exists if from the left and from the right we approach the same value on our function now what about this what has to happen for a limit to exist if we have more than one independent variable well well there's there's an issue right the issue is a very huge ambiguity right now are we traveling along one axis of one indefinite variable the answer is no no what we need to happen is that that order you remember doing this from last section if if you know you got to go back and kind of remember this but uh what we're doing here is we're plugging in ordered pair that ordered pair is along the X Y plane we're plugging in that pair that order parents give us a height above the point on the XY plane does that make sense so we have to Traverse This Plane we Traverse the plane now see what's the height doing as we're traversing This Plane well that means instead of approaching one number on one axis we're approaching one point but we're doing it from what what from what from what direction how are we we doing this what is the curve in this case this is the whole situation how do we know this limit exists well if it equals L done y that does that tell you nothing it tells you nothing here's the issue and we we just covered part of it what's that give you what's a function in one variable what's it give you come on gives you a curve what you what you said on a on a plane this is just a curve on a plane that's all it is what does and this is the whole reason why I had the last section so you get really good at this okay what does a function in two independent variables give you curve it's not a curve that's the whole point if it was a curve it'd be easy we'd have Cal one again done one variable you need how many dimensions to graphic two ah don't make me drop this one dimension okay I do it uh roing like it'sot two variables how many dimensions do you need to graph a function with two independent variables that means it's a surface surface you remember just talking about this you go to a point get a high my voice is getting squeaky come on don't do that to me you go to a point on the plane you get a height a lot of different heights creates a surface for you what we're talking about on this situation is what's the height of the function doing as I'm allowing my point to get close to here here but there's a big problem and here's the problem service Point other point how you feel all the time in this class just kidding let's say that this is my point all right this is my point right here this is the point a or well the height above the point so you got to imagine that this is the height the the whole idea here the point is that this point is on the surface here's my question if the limit here says and this is the whole reason why we did it you have to travel not along the axis but along the curve what's the height of the curve doing from both directions what's the height of the curve doing from both directions well that gives you one path you see on on a curve there there's a certain path there's only two ways you can get to this value from the left and from the right that's two ways if they equal from the left and from the right we say that that's it that's the only two ways you can get there there's no magic Third Way coming from Imaginary Land all right isn't Harry Potter with mathan done that's not what we're talking about there's literally only two ways so when we're doing this we say okay for one variable it's kind of comfortable I think I'm going to have a balloon all the time just because to rest my arm right there perfect for one variable this is a curve it's a curve it's a certain path there's only one path and what that means the whole important thing about this the reason why we were able to find limit so easily in Cal 1 is there are only two directions you can take to get to this value you can go from the left and you can go from the right there's only two directions you get you can take to this value from the left and from the right so that limits us to two directions I'm going to say Point uh what I mean is the value I'm going to say point just so it makes a little more sense for this but but here's the whole idea with one variable you have one path you have a certain path there's only two ways you can approach that value that point from the left from the right so hands understand the concept here's the idea now you've got a surface I say tell you what I want you this is the function all right this is what this means you already told me that two independent variables creates a surface Let's Pretend This is it and I want to let my points my things that are creating this height get to that value how many ways can I approach that this is the surface right which means that my my function is traveling along this how many paths can I take to get here imagine this is a Mountaintop and here's a little cabin that's a horrible cabin and you say hey I want to get I want to get to my cabin and there's no Road pick any road you want how many roads can you take to your cabin I can do this right I can take that road or I could take this road or I could I could go this way or I can start over here and go I I like Smiley I Smiley I'm not drunk and I go what back over here here's my question how many paths path along your surface how many paths along your surface could possibly get you to that point how many infinite that blows this thing out of the water there aren't just two paths anymore curve did you guys get the idea that curves are two paths left right done on surfaces how many P why I completely kind of functioned that thing up didn't I really it up um so how many paths can you take to get to a point an infinite Min you you could go left doesn't make sense right doesn't be up down you could go infinitely many paths to get to this point on a Surface do you guys understand the idea let's write out a couple of those ideas and then I'll show you what we can do what we can't do so for two variables it's no longer a curve this function is a surface and the problem is there's an infinite number of paths along that surface that could intersect that point that could get to that that value on the function for I know I've I've talked it for a while I said I was going to talk about it for about 10 minutes just so you understand the idea 10 it's going to take another 10 minutes before we get to a well maybe not 10 10 to do our first example and why we're doing that but do you understand the major major problem between one variable and two or more variables instead of one where you have only two directions now we have an infinite number of paths that's getting us to our point infinite number here's the problem if you wanted to prove that the limit exists on this surface at that value you have to prove that along all of those paths they get to that point is that easy to do do you want to write out all of the paths you literally can't because there's infinite many of them there's infinite many Vector functions curves that are on the surface that go here infinite many right so how do we do it well it's really hard to do uh you should read that impossible in most cases to do that we can do it with something called The Squeeze theorem calc one concept we're going to use it again all right we'll do it later it's it's hard it's not that easy to do I'll show you how you can prove that limits exist with the squeeze theorem do we generally do that not really here's why it's a lot easier to do two things one it's a lot easier to prove limits don't exist it's really easy uh left side doesn't meet right side or One path doesn't meet the other path it only takes one example and you can prove a limit doesn't exist you guys understand the idea to prove something that's true you got to prove for all the cases to prove something that's not true just got to unpro it for one case done so a couple things about this and we're going to we're going to get through it here in a second so to prove that this exists you have to prove that along all the paths we approach the same height as we approach that point all paths all Infinity of them we get a prove that along all paths we approach the same point really hard squeeze theem would do it we can use squeeze theem I'm going to come back I'm going to prove that but that's going to be the end but it's it's not easy in general sometimes it's sometimes it's impossible we we just can't do it so how do we prove these things exist one of them is squeeze theorem the other way is that we use a lot of the principles about functions that we know for instance we're going to talk about continuity you remember one if you have a function that's continuous the limit has to exist at every single point do you remember talking about that say hey it's continuous here continuous no jumps no gaps no holes no no whatevers uh no nothing that makes it undefined the limit's going to exist at every single one of those points and it will equal the function value of those points we can use stuff like that so we're going to use continuity to help us out with some of these limits to let us evalue them really quickly that's pretty nice so we can do that so squeeze thems one way using some of the properties of of limits from Cal On's another way other than that other than that we can easily relatively easy prove when a limit doesn't exist it's really akin to to what we're doing here from One Direction and from another Direction along the curve we're equal in a different value that that that's that's the idea on a Surface it's from One path and another path we'd equal a different value I can't show you with this because this is going and we're going to show this uh a little bit later this will be continuous at any path because it's defined everywhere there's no holes there's no ASM tootes in 3D those look weird uh but there's nothing like that ASM totic planes there's nothing like that so this will be continuous everywhere which means that all the paths lead to this that one point at that point so that's okay but in general it says this says just find me two paths that when we approach my point XY it gives me a different height I'm going say that one more time so it sinks in because some of you guys are zoning out already all right I'm going say this find me two paths that when I get to my XY when I get to this it gives me out two different heights that is what this situation would be in 3D you go I along One path I'm up here along another path same surface I'm down here okay that right there tells you you're not continuous at that point and therefore the limit does not exist at that point that's what we're doing and it only takes two paths because if we can show two paths don't equal the same height that right there is enough to say the limit doesn't exist why because for the limit to exist we have to show that every single path goes to the same value if we prove that two don't we've proven that wrong that's the point so we can easily prove a limit does not exist by showing that along two paths we get a different value as we approach the same point they feel approach wrong like 14 different times I did sorry I don't even know which one is anymore you ever do that like you know it's one way but you write it two different ways and you can't tell anymore I'm pretty sure these two are wrong down there don't ignore those so white them out on the on the whatever draw it in there typical math teacher right typical this should even let me write words iore M anymore what was I approach thanks that's really help for we get a different value the different value is the height of your function it's on the surface F of XY as we approach the same point XY that's a horrible XY I really don't feel like explaining this anymore have explained I know if you guys understand the main Concepts one variable easy don't exist done one variable easy exist for two variables not easy you have to prove every every path goes the same has the same value not easy to do to prove it doesn't exist it is easy just Pro prove at least two at least two paths have different heights when we are getting to the same x value so here x y what's going on from one way we're up here from another way we're down here those are different heights limit does not exist because we fail that show F feel okay with that one let's do a quick example um some of these are very easy uh very easy to do I'm going to show you exactly how to you go about telling what you should do and when you should do it you know just to kind of recap a couple things ladies and gentlemen how many independent variables do you see right now very good everybody come on how many independent variables what would this graph look like 2D 3D 4D this is a Surface in 3D what I'm asking you is does the limit exists on this function so is the height of this function so the surface continuous at that point do do we have continuity do we have the limit existing from every path can we go the same place I don't know what I'm saying is at this particular Point as X and Y the point approaches 0 0 from what direction it's on a plane ladies and gentlemen there are no directions any direction you could get to 0 on the plane you think about that way too here's a plane here's 0 the origin on the plane how could you get there infinite number of ways there's not just one axis there's Not Just One Direction make sense so I don't like that band One Direction but anyway um it's the wrong direction so can we prove it exists listen I'm going to cheat uh I'm going to tell you some backwards things that we're going to get later but you use them now so I'm going to teach circularly I don't care anymore uh we're going to I'm going to kind of use something that I'm going to prove later and then I'm going to use that thing to prove it later so it's it's going to be a little circular uh here's the deal because I just want you to be able to do it pretty easily and we're going to use continuity to do it the first thing you should do every single time when you get a limit from back back from Cal one is what plug it in just plug it in just see if it actually works and we're going to use that later so try to plug it in now what do you get 0 over 0 loby talls oh crap can we do ly talls do you know how to take derivatives with two independent variables yes you do how oh crap you have you have physics don't you suckers not yet next section yeah changed a little bit it's weird um but no because what would you do with your partials right partial deriva is what what are you going to do with that as far as your limits concerned all that would do because all you know right now from your physics class is that there are these things called partial derivatives some of you do not even know what they mean yet you just know that you have to do them and this is how you do them what they will do for you is they will give you a path that is over a certain axis that's all they do they limit your travel along a surface over a certain axis I will explain that in next section so will they help us they could but not generally no what we do is first you try to plug in you get 0 over Z you okay well that's a problem if it worked out to a number that's defined done you're you're done I'll show you why later we'll talk about continuity later but you'd be done here's what we're going to try to do now what I'm going to ask you specific specifically for is show that this limit does not exist not try to prove that it does exist because it's not going to but show that it doesn't exist how how how many paths do we need to prove a limit exists infinite how many paths do we need to prove a limit doesn't exist so let's pick two paths I I'll tell you something right now always pick X or Y = 0 as one of them that's the easiest path now I'll I'll show you that so we're going to put a long along x = 0 oh man I need another balloon along xal 0 here's what this does can you imagine can you imagine this so along x = 0 imagine this at the origin with that point is right right above the Z okay so this is the axis here's what's happening if I let you do x = 0 xal 0 goes way back here then what's the only variable that we have left cuz I asked you oh that was funny come on yeah it's y I know if I let if I let x equals z then I say let's let's force that to be zero we would be traveling along the Y AIS please notice something right now I'll explain it a little bit later as well I'll reiterate it but notice something 0 0 is along the Y AIS do you see that we have to whatever we restrict it's got to be a solution have that that point as as a point on that that line we have to do that we can't go I'm going to go over here we got to travel over whatever we're restricting this to so if I say let's let x equals z then what this implies is that I'm going to be approaching that point the point down down here here's the function value but only along the Y Direction so this would be the curve that's doing this say I'm going to take the path where X is zero that's right along I'm going to try to do it this way so you see it right along just the y direction just right over or under the y axis do you guys get what I'm talking about so if I let this go along xal 0 what about the Z the Z is the height I don't have to worry about that that's a dependent variable I'm only worried independent so if I let x equals z what I'm doing I'm traveling over the surface right over the Y AIS did you guys get that so this right here is traveling along the surface directly over the Y AIS not the X xal 0 is not over the xaxis xal 0 is over the y- axis now let's see what it does mathematically if I force x to be zero look look stop writing just look for a second okay if I force x to be zero what happens to my x's they are zero the only variable that now changes is the Y variable that changes our limit completely if I go along xal 0 all my x's are zero this limit now becomes a limit of what's on the numerator if x is zero well you know what we'll do it how much is X as but wait wait for it as what approaches what is X approaching anything anymore no we don't even need it because X is fixed so what that allows us to do is basically take a two sorry 3D limit and change into a 2d limit or take two independent variables and make it let's see which one this one make it one independent variable so instead of X Y approaching 0 0 forget it man I'm already stuck here I've already let X go to zero I'm just along that path of course it's going to hit x equals 0 at some Val at some time at some somewhere along my Surface but now I'm only letting the Y change does that make sense to you if I'm along the Y man I hope it's sinking in if I'm letting along xal 0 that's literally solely above the y axis do you guys get it which means that the only thing that can be changing is the Y I'm right over that that's it let let's make it easy y^2 over y^2 um I don't care how fast your slow you do your limits at this point once we change it here you're you're done if you want to match degrees which is always a great idea you can do it if you want to do lby talls two times fine divide by large power denominator I don't care uh what are you going to get out of this thing perfect that was Heaven saying I love these answers uh no it's just Adam's phone it's okay uh question did that prove the limit exists have we shown every possible path we' shown one the one that's right let's talk about every possible I don't have infinite time to do that uh we have one path right here all we have to do is show One path doesn't exist all right it's literally the point uh show me one other path along what what do you think intuitively what do you think let's try that why don't you set that up as you're writing this I'm just say a little blur along y equals z shuts down the Y AIS it forces you to only travel the surface right above your only other independent variable which is X so this would be traveling the surface right along the x axis that's it it would give you this so if this is the Y the X would be traveling this way so traveling on the surface but right above the x-axis it'd be trying to give you this idea now what I'm going to try to show you you here is that these things don't match up on this graph or on this balloon they would match up this would not be good for us I'll show you what to do in that situation just a minute did you at least uh let your yv value go to zero if you do then this is zero this is zero these are still X's but since you let y equals z you're right above the xaxis the xais is your only legitimate independent variable left you're saying if I'm over the x- axis the x axis is only the X variable is the only thing that can change so I don't have this anymore I just let X approach do you see why X would still be approaching zero though I still need to make it to that point I've already taken y equal 0 so it's going to whatever path I do is going to make it through that y equal 0 value but I also want X to go to zero because I need to make it to zero zero those two ideas have to be kind of cohesive does that make sense to you I just did a whole lot of words and so we like uh if you let y equals z already and change this all Al so it's only x^2 - 0 okay done over 2x^2 + 0 done you're trying to get to that point correct you have to make it to that point if y equals 0 I now need to say okay I know I know my Y is z right over X I need to let my x coordinate approach that zero value so that I hit 0 0 make sense we just do it one at a time uh how much is that you have just shown this situation on a Surface it's practically the same it's just that from this way and from this way you're getting two different heights there's infinite paths but I you show along the surface along two curves two paths that I get different heights at one point just above one point and that's exactly what I've done so by showing that along two paths what paths right over the y- AIS right over the x-axis along those two paths I have a different value for my Surface and that says the limit does not exist because along all paths we' have to come to the same value for our function same value for our surface show pant be okay with that one so we'll wrap this up here so you show it in your test you say um hey -1 does not equal 12 therefore the limit does not exist please be very careful on your verbiage here the limit doesn't exist at what point 0 0 at the input 0 0 other points sure it's going to exist most other points can I give you a couple hints that's going to really help you out here's the quickest easiest simplest way to do limits or to show the limits don't exist but you you got to listen carefully because I'm not going to write a whole I'll write a couple things out but not a lot check it out ready ready cover up your wise do the limb in your head you get one half you see it cover up your X's do Li in your head you get netive one did you catch that if those numbers are different go along xal 0 and yal 0 done because that's exactly what we did we let the y equal 0 1 half we let the xal 0a 1 different values different height done just go just show your work go along xal 0 yal 0 and you're you're done does that make sense to you that that's the easiest way to go about it the problem is what happens when we go along x equals z and y equals z and we get the same value still and that's that's the nature of the next couple couple questions that we have um so a few notes and we continue so number one try xal 0 and yal 0 first in your in your head don't even write them out yet just in your head just cover up the variables and see if you get different numbers if you do then only xal 0 y equal 0 that that's it that's all you got to do so try these paths first the reason why is because you can do them in your head very very fast that's why listen carefully this is not going to work all the time if it doesn't work so it means you do x equals 0 and y equal 0 you get the same value still have you proved something exists have you proved something doesn't exist no the that's the worst case scenario then you go okay well what do I do now try another path but always use at least always use one of these to to do one path okay so always use one of those if these don't work choose other paths a few notes Here there there's a couple that I want I want you to know that that are really going to help you U number one Whatever path you choose must contain that point that has to be the cas why would go okay I want to I want to P I want to check the limit at 0 so let's go through 52 yeah does that make sense no you want you want to make sure that path actually contains that point so number one thing make sure the path you're choosing contains that point does xal 0 at some point contain 0 0 yeah when X is z and then when y becomes zero that's why you got to go along y okay so make sure that happens for any substitution you make any any path you choose be certain your point is actually on your path second when you're doing this it often helps to substitute so that the degrees on your numerator and denominator become the same and thirdly for one of your paths always choose one of these always choose one of those because it's the easiest one we can do so always pick one of those if you can almost 99% of time you can It generally does not matter which one you do I would like to try uh just three examples we'll talk about um having more than more than two variables we go to three variables after that we'll talk about continuity and that wraps this section up sounds like only a little bit it's going to take us a little while because I want to fully explain this again limits as you know from calan right limits were one of the fallbacks for people they go I hate limits why because I don't understand them I want to make sure you actually understand this do you understand so far what we've been doing for real don't just nod your head and go yeah I get it do you actually get the idea of what we're doing yes the over the surface uh sorry along the surface over a particular path you guys get that okay so let's try this let's do the limit here there everyone in class right now how many independent variables do you have okay so this is going to be a what type of surface it's a surface 3D surface great what point are we approaching what should you do every single time with a limit no matter what I don't care what class you're in what what do you do with the limit just try to plug it in what are you going to get here okay well that kind of sucks so we can't just plug in the value because we get Z over zero we don't really know how lal's rule works if it does work don't even know um so goodness gracious what are we going to do well what I'm going to ask you here is not to prove that it does exist prove it doesn't exist that makes your lives easier because all you need is how many paths two paths you can't do one doesn't make sense you got to show that we're getting different values just like from left and right side we're getting different values so we need two paths to go um here's the cover up method that I would encourage you to try first off okay check it out cover up your X's what's the limit there oh silence oh okay you know what we got to use one of those anyway so let's try x equals z let's see what happens can you at least left Siders uh can you tell me what I should rewrite if I'm letting X go to zero tell me please anything you want so just do it tell me what it is oh a limit oh good I still have those good uh What variables do I have why why where's it going zero can you verify something for me that this 0 is somewhere along the Y AIS which is x0 can you verify that for me it's somewhere along there you have to have that happen if it doesn't happen you're you're wasting your time um what happens when I let X go to zero over what what is that it's zero two ways you can prove it one oh wait wait how many independent variables do you have now one does lal's rule work yeah now it does easily you get 0 over two if you want to do L spell rule twice fine or you just go well that's an actual zero this is only approaching zero so it's 0 divided by an actual number not zero keep in mind what limits do they say I'm getting really close so that's zero for sure heading on if you're okay with that one you can do that by just going um hey uh let's let's cover this up can you still do Lal rule yeah you get 1 over 2 Y which is 0 over 2 it's zero now try this what if I cover up the should be on this side what if I cover up the Y let y equal Z do I still get zero yes that's a problem and I'll show you right now if I go along yal 0 then I'm fixing the Y I still need the X to go to zero notice 0 0 is along that path if I let y go to zero then I still get zero but I have 3x2 you guys see where the 3x2 came from where the zero came from yes or no how much is that limit now if you didn't get it the first time let's get it this time I got zero along this path I got zero along this path does it prove the limit exists how many paths can get to this point infinitely many it's a it's the origin on a plane infinitely many lines paths curves whatever get to that value I've shown two that exist have I proven that the limit exists have I proven limit doesn't exist which is my question anyway I've done nothing I've done nothing so far now not nothing one of the these is good okay this is at least one path now both of them are you don't need both but choose one of these CU it's really easy to find so either one I don't care but it's not enough verify that this is not enough for me please let's pick another path here's how to pick the other path first be certain the point you pick is actually on your path we're going to do that substitute so that the oh this is the big one so that the degrees of the numerator and denominator are equal and always use e okay we already got that way down so here's the other path I would choose I'm going to look over here and I'm going to choose you know what I'll make it look a little side I'm going to choose so that this happens I want to pick yals a function of X or x = a function of Y I'm going to pick that but I want to do it so that these conditions are met so I still have this as a solution that's got to be there and so that my degrees are are are equal um there's multiple ways to do it sometimes but here's the one I'm going to pick because I like setting y equal to X's you guys get that so so check this out let's imagine something let's imagine if this y was actually an X could you multiply those yes what would you get if this y was actually an X could you add those and you would get 4 X2 do you see how I'd be matching up the degree of the numerator denominator that's real nice that's what I'm I'm looking for also you so it sinks in so you get it I I probably don't have to explain it anymore but just so it sinks in notice something if you were to graph yal x look if you were to graph y x is that a curve yeah well okay we call lines curves is that a curve does it go through that point can you make a substitution if you did you go okay uh well what's the only variable remaining now if I let y = x I have a limit I have 3x * what am I letting y equal notice how you could do the other way around you could let xal y it doesn't matter but just change it so you only have one variable why well because we know how to deal with limits with one variable does that make sense so oops let yal X let yal X seriously though are you guys okay with that one now if we're only in terms of one variable I really don't need to let y approach anything I let y equal x that's fixing the y That's fixing it to the x value it's making it now depend on the X which means the only thing I let float around is my X and it still has to go to zero they still got to approach that same value for real yes no are you picking up what I'm turn to yes now let's uh let's do this so here if I change this make it a little bit nicer I got a limit 3x2 we're not going to waste time doing limits real real slow what's that limit by a variety of techniques now here's the question does that prove that the limit of this function at that point does not exist does that prove it along One path we got to zero along another path we get to that didn't prove anything along a different path we get to 3/4s since 0er and choose either one I don't care since 0 does not equal 3/4s that proves that the limit does not exist at x equals or as the function approaches 0 0 on the XY Point tell you what I wanted I'd like to do two more since we're right on this um we only they're going to go a little bit quicker um since we're right in the middle of this this concept and when we come back from our break I'll talk about how to deal with three variables it's actually it can be a little bit easier because you don't have real nasty substitutions and you'll have this idea down are you guys ready for this so let's try uh let's try one more to get you know what tell you what I want you to try this one on your own at least the first couple steps of it let's see so as I'm a racing here's what I would like for you to do I'd like for you to go along the two paths that we pretty much always start with okay well actually I want you plug in the number what are you going to get z z over zero so that means okay I want you to prove that this doesn't exist right now how I want you to start these I want you to either in your head or write out these paths if you can do it in your head do it in your head but you got to write at least one of them you always have one of those paths so I'd write this one out and i' I'd go with xal 0 and see what happens then in your head or write the yal 01 if you got different numbers you're done if you don't then you got to pick a different path that's the hard part here so I want you to try the xal 0 y equal 0 thing see what you get and then work on finding a different path e let's just stop for a second I know you're still working I know you're probably thinking about that path right now at least I hope you've made it that far so so far let let's think about this if I let xal 0 I'm traveling along the Y AIS 0 0 is on the Y AIS that's great I also know that zero x if x is z that's zero the whole thing's zero gone if that's zero I get y the 6 and this is SLE if I let y equals z the only independent variable left is X it still has to approach Z I know that y 0 still contains that point I know that y that's 0 that's 0 we get 2x^2 and that's 0 that does nothing for you that gives you about one point out of 10 on a test because what have you done nothing literally nothing you haven't done anything you haven't proved it exists you haven't proved it doesn't exist you done nothing so along let's pick another Point let's do this let's try to pick a substitution that not only contains this point but makes the degrees the same if we can that's what I want to do so so let's let's see how to do that uh if we're going to try to make the degrees the same here's a little hint I'm not going to write down maybe you should write it down um take the the variable with the smaller degree and try to make it into the variable with a larger degree otherwise it's kind of hard because if I want to change well what's what's got the larger degrees here X's or y's y's do if I try to change y's into X's I'm going to have to cut powers does that make sense that's hard it's easier to grow Powers so if I say I want X to equal something in y such that the variables have the same degree can you see the substitution right now yes what is it good so I'm going to take the smaller degree variable and try to change it into having that power so if I say I want X to be equal to you guys get the concept that I'm going have to eliminate a variable right CU I can't do this stuff in two variables very easily so I'm going have to eliminate one so I'm going to change X's into y's or y's into X's I also want to do it so that my degrees become the same because the the limits are way easier that way the limits you just basically cover them up so I'm going to say hey let's go along the path x = y 3 do you see why we might want to do that if I let X if this is a y the 3 that's y 6th if that's a y the 3 that's y the 6 and I can add them and then it's very nice very easy your feel okay with that one also one more thing you got to double check this is 0 0 on that curve notice it is actually a curve right now it's not a line it's a curve through space is it on that curve yes 0 equals 0 done so that says that if we let all X's change to y's y 3 then this is y 3 * y 3 * cosine y 3 X was y 3r now X is y 3r now if x is y 3r I get Y 3r 2 + y 6 uh oops what's the only independent variable left so let's let that y go to where it's supposed to go to for real show hands feel okay with that one it's kind of a weird one are you guys back there okay with it are you sure right side I don't want to lose you okay we spent too much time doing this for me to just lose you now this should be the kind of kind of funner stuff interestinger stuff it's a word can you simplify it let's try how much this is 2 y 6 + 1 y 6 we get 3 y 6 with me yeah know can you simplify that ladies and gentlemen this is why we try to make the degrees the same because if we can we can do some basic basic math operations gone get them gone that's great oh my gosh sorry why why why do you people let me do that goodness gracious I just like threes today I wrote so many of them that's why so we left with cine y 3r over 3 what do you do every time in caline when you do anything with a limit what do you try over and over and over again what should you try now PL plug it in hey what's what's y 3 if Y is z what's cosine of0 does that prove anything okay everyone what's it prove why why because along those two paths this one a curve along the XY plane which we're floating on this this is weird right this is a this like S curve so we're we're floating on the surface above that S curve and we're getting to this this value that point we're getting to that value and then we're right above either one of these axes and we're getting to a different value so the height of the surface is a different value as I'm approaching along that along the surface over those two curves so since 0 does not equal 1/3 limit does not exist at that yeah yes no okay tell you what one more I'm going to I'm basically just going to do this one for you I'm going to talk through it just to show you some weird things that can happen because this doesn't tell the whole story we're not always going to z0 your substitution is not always going to be xal y or yal X it's not how it is all the time so I want to do one more at least uh to give you that there we go what's the first thing you do with limits like all the time if we plug that in for x = 1 and y = 0 do you see maybe you can think about it for a second that we're going to get 0 over 0 x is 1 so that's well on the numerator you get 0 0 but down here we got 0 we got 1 - 2 + 1 that's 0 over Z if it wasn't you'd be done plug it in you'd be done we'll talk about why later but but you'd be done just plug it in so since we get 0 over Z we go well let's show this limit doesn't exist or let's try everyone in class right now at what path do you use at least what path do you do you use you use at least one of these all the time what what is it well either one either X is zero or Y is z so so pick one um in this case I I think I did y equals z so if y equals z we got limit oh wait now wait a second if we're going along yal 0 this is zero and that's zero that's true this is zero I get x^2 + I get that can you tell me where the X should be approaching is it true that along x = 0 I'm going to hit X = I'm going to hit this point 1 Z notice that yal 0 is the xaxis do you guys get that so that point is on the xaxis one Z it's on there so I'm going to be going through that point but you do got to make sure you go to the right value so X still has to approach one since I'm fixing y equals 0 x still has approach one CH you okay with that one okay so what's it give you Z it gives you zero not Z over Z it gives you zero uh because if you do lowy talls or whatever you have you actually do get the the zero out of that now if I go along let's just see if you're paying attention should I do it should I what not on that curve that Point's not even on that curve does that right there go through that point no then don't do it that's still you're wasting your time you're wasting your time what about this would that be something I could do yeah that that goes through that point that's fine now let's just try to our heads okay you don't have to write it out but try in your heads if you did if x equal oh you know what fine write it out out here why would have to go to zero if x is one we get that because we have 1 - 2 + 1 You' get that the only independent variable left is y y has to approach zero I got one here so we get what's that limit come on does it do anything so we need another path now sometimes to get your path because right now it's not obvious you go man I have no idea what path I should take because if I go uh trying to match up variables I start having ones here that doesn't that doesn't really help me if I let X = Y that doesn't really help me I get some squares do you guys see the problem I'm talking about are you sure so here we go y equals z done no problem X approaches one got to go to the point uh xals one got to go through the point y's got to approach zero either way get z0 it's not telling us it's not telling us anything Sometimes some tricks like this help not tricks but some simple math try factoring see if you could Factor this stuff sometimes that gives you a path to go along for instance if I factor and if I factor this pull the y^2 but notice how x^2 - 2x + 1 that's still here that's a perfect square trinomial in fact it's x -1 squar oh shoot hope you saw the little change there you guys okay with that one it should be a plus should be a plus because you have y^ 2 plus this this is that piece you should have a plus times does that make sense to you I know it's just algebra just algebra is what messes you up so be careful with that should F okay with that one for real okay now if you have that you go well well how about how about this is there a path that I can go along that's going to make my degrees the same notice I have a power two I've got a power two I've got a power one I want a power two and these are the same go along that path let's change a whole piece of it let's change that ugly piece if we change that ugly piece we go hey let's just change X-1 into a y h now if you're okay with that one now stop right now stop stop let's see is this point actually on that path plug in the one for the X plug in the zero for the Y does it equal yes 0 equal 0 you know that Point's on that path does that make sense it's going to work so let's see we'd have limit let's talk about an independent variable in a second am I still going to have a 2 y I don't want to replace the Y's okay I want to replace this am I still going to have a 2 y am I still going to have an x - one no what am I going to have y this is a y^2 plus if this is a y that's also y s what's the only independent variable I have left come on folks quickly and what's it going to zero CU X we we fixed X along this path so we said along the surface over this path and now we're letting y go to zero the x is done X is tied up in the Y's we don't have to worry about it it's just this independent variable that's moving now we did one big fat one question did it prove anything yes come on put it together why what did it prove does not exist why what's it mean why does it I know that zero doal one one does equal zero you're saying the same thing cuz at that point they'll have different heights because what at that point they'll have different heights different heights all I'm doing along these two paths different heights that's it limit doesn't exist so 0 does not equal 1 because these are the heights of the function along those two path along the the surface over those two paths so along the surface over those two paths the function has different therefore the limit does not exist at that certain Val do you feel like you fully understand this concept up to this point because limits can be tricky uh but I've given you some good techniques here we've shown limits don't how to show limits don't exist when we come back I'll do uh three variables we'll talk about how to do that with with three variables we'll talk about continuity very quickly um and then we'll talk I'll do the squeeze theem and show you some pretty pictures all right so how do we do this when you have more than two independent variables we made the transition from one to two one was is easy Cal one two I showed you how to prove limits don't exist uh with two now we'll talk about what happens when you have functions with three count them three independent variables what type of a surface is that four dimension four dimensions now that's weird because the paths now are in three dimensions does that make sense so we're traveling along p here we were oh sorry I rushed it all um [Music] in in one independent variable you were along a curve basically you only had one dimension of of travel left right that's it with two independent variables you had two dimensions of travel we onun a plane and then surface said you're restricted to these two-dimensional curves that that were going that were traveling above with three independent variables are curves they are still curves but they're space curves they're now in 3D well what's nice it's actually nice it's not hard what's nice don't overthink this we're not traveling a long uh surfaces we're still traveling long curves over these 40 surfaces but for three variables the paths are now parametric because that's how we defined Curves in 3D they're now parametric does that make sense to you and that's we're going to be traveling over paths in 3D that are parametric curves quickly everyone what variable do we use for parametric curves so your whole goal in life right now is to replace X Y and Z with t that's literally all we're going to be doing for for one of these things that's it we're going to be traveling along those paths now this one this is this is nice some people don't talk about this talk about this always choose an axis as one of your paths always choose an axis as one of your paths because it's really easy to do that so I want to show you how to choose an axis then we'll talk about how parametrics play A Part in this because otherwise it it's really confusing so let's let's start this so the first thing I want you to do is go along and access what axis do you want to go along x x love it let's go long X I want to explain what's going on we are in four the curve the surf sorry not the curve the surface is in 4D but we are traveling along that surface over paths in 3D just like in 3D we're traveling over paths in 2D do you guys get it so we're going to be traveling along this weird 4D surface over a 3dimensional curve that's what we're trying to describe right now one threedimensional curve is simply the x-axis that is a curve in 3D is it a boring one but how do we describe it well if we're talking about X's y's and Z's how do we get the x-axis what what's the Y value for anything along the x-axis what is it zero and what's a z value for anything along the x- axis Z so it's really easy just set X sorry set u y and z both equal to zero that's probably the easiest thing we've done all day just set both x and y equal to zero and you will be along the x-axis head now that you understand that so what we're looking at right now is what's the value of the function the 40 surface as it's traveling over the xais a 3D curve through space that's what we're doing so if we do that the limit is oh well um hey if we let yal 0 and zal 0 we have Z so that's 0 0 0 0 0 over what and what's our only independent variable left ladies and gentlemen and what's it got to approach how much is this limit zero perfect that's why we always choose an axis first if we can that's why we choose an axis first because it's really really easy you just set Y and equal to Zer or X and any two of them and you're traveling along an axis should F okay with that one you can choose z-axis here because Z is now an independent variable unlike before okay it's independent so we can do that you guys understand the concept of doing this weird concept of we're now on a 4D surface and the path that we're over is a 3D path so what's the other one well the other one we're going to have to go along a curve that's parametric the biggest piece of advice I can tell you is set all of your VAR ibles equal to a function of T CU that's a parameter such that your degrees match up just like we've practiced so we're going to go along a curve call it C go along a curve where well let's look at it I'm going to replace X's y's and Z's with T's what's going to let the powers match up well what I'm looking at is this is x^2 y^2 and z^2 if X Y and Z all become become the same variable will I be able to combine that stuff let's just let X and Y and Z all equal t x = t y = t z t what is it what is xal T yal T and zal C what is that what is that curve line it's a line through space that's all it is it's a do you see the parametric line that's what's that's what we're traveling over now so in 4D we're going to be traveling along the surface over that 3 dimensional line that's it that's why we use parametrics because to travel three space we need parametrics so F be okay with that one now let's make the substitution if x and y and z are all T then our limit becomes well let's let's see now um H TT TT TT t² t² t² super fun uh hey what now now this is an interesting question what is now our independent variable since we basically just parameterized this function T what's T approaching Z not 0 0 0 CU if we let watch carefully if we let T approach Z X and Y and Z also approach zero so letting T approach zero lets the the the Point approaches 0 0 0 does that make sense so just got to watch that sort of stuff so T is approaching Z so that's the limit as T approaches zero of t^2 t^2 let's see 3 t^2 over 3 t^2 how much does that equal come does that prove something for us yeah so that be 3^2 over 3^2 oh yeah it's one it wouldn't have been one before sorry what does it prove for us that it doesn't why does not exist because as I'm traveling along the 4D surface over that path the x-axis and over that path which is a line through space the value of the function is different that's what we're proving here does it make sense to you so since Z does not one the limit does not exist at 0 0 Please wrap this up at least for my class I want a statement at least something that you actually understand the concept do not just say doesn't exist doesn't exist ever come on we just proved that because of these two paths the limit doesn't exist as we approach that point that's the idea listen I'm not going to make it through the whole thing here I just want to going to give you the blurb about how you you start it don't worry we start up last time um what would you do first if I wanted to prove this doesn't exist what would you what would you do first go along what an axis so you you do maybe the x axis you'd get y = 0 and zal 0 you get 0 over x^2 you'd still get zero do you guys see what I'm talking about what I want to focus more so one of um along X AIS and we would get the limit as X approaches Z 0 over x^2 and that's zero that's still zero you that's an easy one that's nice you always pretty much always do that what I want to focus on is the parametric curve that I want to create this is a little harder because you don't just do X Y and Z all equal the same parameter T what I want you to think about maybe I'm going to give you about 30 seconds of just think just think about it how can I make these Powers all the same how can I make those Powers the same if I can how could I do it so think about it what's the what's the variable with the largest power right now try this try calling the variable with the largest power t let it equal T then try to make the other ones match up so for instance if zal T I would get T to 4th does that make sense I also want this to be t to the 4th what would y have to equal in order to get t to the 4th I also want this to be bless you T 4th uh what would X have to be to get T 4 that would do it that would do it if we do that the limit changes to t 0 for sure and we got t^2 time notice that here's X is t^2 z is T So t^2 plus 2 * oh hey t^2 squar because that's Y 2 then we have t^2 2 t^2 2 and T 4th I guess I am doing the whole thing I always do that I should never say stuff like this can you double check my work I did a lot of it in my head that's right triple check that's right the same person doesn't count when you do it twice did you get the same thing yeah does it prove anything yes zero three fours not the same limit is exist done should fans you okay with with this one now we practice now we practice evaluating limits um I need you to know that evaluating limits is exactly the same here as it is for Cal 1 and I'm going to cheat I told you I'm use some circular reasoning what I'm going to do is show you how to evaluate based on the fact that we have continuity then I'm going to talk about continuity and use that to kind of kind of uh fill in the fact that we're able to do this okay so here's the how I'll take talk about the why in just a in a little while so the the how whenever we're evaluating limits I don't care what class you're in through calculus the first thing I want you to do is plug the numbers in just plug them in if you don't get 0 over Z or a a zero on a denominator you're done plug them in and we're using the fact that if you plug in a value and you get a a number a limit then that function exists at that point and in a neighborhood around it you can find continuity and in a neighborhood around that means all the paths lead to that same value that's what we're using here we'll talk about uh how that how we're if we have a domain that's defined we're continuous on that domain we'll talk about that in a minute but that's basically what we're using we're saying if this function exists if it's defined at that point you're going to be continuous at that point which means that the limit has to exist because if if we're Define the limit exists uh sorry if we're Define and continuous the limit has to exist therefore plug in a number a limit has to equal the value of the function we're kind of cheating here uh the idea from Cal one would be if the point exists and the limit exists and limit equals the point you have continuity if you have this the limit exists the point exists and the points in that little spot that creates continuity so if we have continuity ie it's defined if it's defined we have continuity uh if if we have that then we can go backwards and say if we have continuity then the limit exists and that's what I meant about we're going backwards here so let's try it can you plug in your X and your y-v value what is your x value what's your y value have you already plugged this in how much do you get three okay why well 1 -2 okay perfect one2 plug in you get three simplify you get three you guys okay with that one all right Bo a weird one we also learned this from Cal right side and left side are super fine uh as long as you have continuous functions so honestly plug the numbers in and see if they work so just plug in Z and zero let's see if we can do that can you plug in Z and zero what do you get folks just just this part just that what do you get here so something close to Zero from the right positive something close to Zero from the right also positive you're going to get something close to Zero from the right zero so you're you're going to get zero really close to zero but positive reason why it has to be right sided because you have it under the square root you can't get the from the side okay so you can't have that's that's the only reason why it's there so this is e to 0 over1 how much do we get for that that's it that's it that's evaluated limits head not if you're okay with with that one we can make them nastier but the idea is the same so let's try I'm going to give you one that's kind of nasty and and then I'll give you another one I want you to think about so give it a try here's XY Z I want the limit of the following function as we approach the3 one and the function is oh don't worry worry yeah that's what I'm talking about woo what's the first thing you do with any limit like ever can you just do that just plug it in just plug it in and see what happens hopefully I rot down right that suck after you do that one I want you thinking about the next example please what's the x value y value Z Val instead of trying to do all of this math in your head probably no good idea at least write out some pieces of it all right so so for here I'm writing out e to the S of I I do know that's zero so I I'm going to put the zero e to the S of Z Plus Ln that's got to be there cosine that's got to be there Pi that's got to be there but I would do things like this 3 - 1 is 2 so this is * 2 or you can put 2 pi does that that make sense to you I do just little stuff not a whole bunch it's crazy too hard to do that how much is s of Z how much is cosine of 2 pi how much is that plus equals limit one okay yes no easy medium hard pluging stuff in pretty easy as long as you know to that now did you think about that one as we were we were doing the problem did you think about that one what's the first thing you always do with with limits plug it in what what does it give you crap uh every other time we tried to Pro prove this doesn't exist now we're going to try to prove it does exist there's a few ways that we can go about thinking about this one of a squeeze them which we're not going to do right now we do it a little later a little bit later um one of them squeeze theorem another one is well you can do some algebraic manipulation or you can do a really cool trick sometimes that helps a lot if if you see if you if it's not working for you and you're like dang I don't even know but you see a lot of X2 + Y2 floating around you see a lot of that you see what I'm talking about number one thing goodness don't make me drink at night please don't start canceling this stuff out I like Crossing stuffff out I just cross it out please don't do that I swear please I'm begging begging please don't do that okay also I don't want you to start factoring you can't Factor this it's impossible you can factor that but it's not going to help you you get your X Plus y times this quadratic that's not factorable so you go what I'm do with that that's that's not going to help what I do see is that goodness um I know that this is true as long as they let right this and that also be true sometimes we can trick our problem into thinking that it's polar make it polar if you get Z over you see a lot of X2 + y^ 0 0 you go well let let's try to make this thing happen if we do we can let this go to R well X would be R cosine Theta so R Cub cosine Cub thet plus r Cub sin Cub th he now if you're okay with with that one for real yes no yeah you see where this came from if x is R cosine Theta then X Cub is R cosine Theta Cub R Cub Cosa Cub same thing for the Y what's the denominator what's it going to be my next question here this is a really kind of interesting question what's my independent variable what's it going what's what's X and Y relate to up here say what X and Y are only tied well they're tied to Thea but they're also tied to tied to R well let's see what happens if X and Y go to 0 0 what's x^2 + y^2 going to then R would also go to zero but look it can't be negative all right this is going to give us something that's positive so plus right hand side so if we're going to use that then our R is a thing that's going to zero but it's right side now can you see something nice to do come on quickly see something nice to do what are you going to do not yet what are you going to do simplify you're going to simplify how it's the f word come on what are you going to do so if we Factor and we simplify let's think about it I don't even care what this is tell me why I don't even care about that what that is what's the r doing Z what's Z time pretty much anything and even if it's zero I don't I don't really care so 0 time anything is how much Z zero does that prove the limit exists yes yes it does we plugged in the number we actually got a number out that proves the limit exists head now if you're okay with with that one question why does the r have to go or zero from the right again I missed that well couple reasons um when man you could go absolute value you could go absolute value but you're never going to get a negative out of that the absolute value of R if you took the square root of both sides uh you you'd get the absolute value of that R we just go Zer plus to not have to do with a negative honestly so that's why not a very VAP thing just make sure you go to zero can you do it for the next one can you see the substitution for the next problem cuz right now if you plug this in do you know what you get if you plug that in plug in Z plug in 0 0 what do you what do you get come on folks how much is this times how much is this this is zero Ln of 0 is negative Infinity Ln as you approach Z negative it's doing this that's what's happening Ln of Z negative Infinity this is an indeterminate form if only in determinate form that's what that is well if we start doing stuff like all right let's let X2 + y^2 = r 2 then we can change our limit to R Ln r s quick head now if you're okay with the substitution please don't let your R squ change to RS magically don't do that X2 + y^2 is R 2 are you following the math on this you see why our R is going to go to zero now that doesn't help you with the indeterminate form but if you ever did limits of indeterminate forms from section 6.7 from calculus 2 then you know that at this point if you have zero times one some type of infinity and you create this complex fraction you change the zero into a different you change it into a different thing you change it into Infinity now you have Infinity notice this what's one over zero oh it's Infinity what do you do anytime you see Infinity over infinity come on notice say what so here even if you move this down lals how you going to do it because you got two variables two independent variables how you going to do that but if we change it to one independent variable and we manipulate it now you got lals I'm not going to finish it I want you to finish it try lals here so kind of a cool idea I hope you like this idea I liked it the first time I saw I thought was pretty neat you go what am I supposed to do with this I no give up no let's try to change it to Polar let's make X2 + y^2 = R2 we got some substitutions for X and Y same thing here as soon as we do that even if we have indeterminant forms you know what to do with one variable you can start creating from zero time infinity infinity over change it make a fraction over infinity and then lits head not if you're okay with that one for real comments questions anything I can't help you if you don't ask yeah okay anybody else yeah how do you get one R you force this thing down to a denominator you force it so you go well where where do I want that I want to create this get form it's already an indeterminate form what you're forcing it to be is a form that allows lals so you go how do I move this to a denominator well uh you could do R -2 that forces it right y know about negative exponents okay well what is r the -2 well it's actually a complex fraction now you now it's Infinity over infinity you can't do anything with this please I'm begging you don't go zero time Infinity zero sometimes that's D for done right there though D for done okay sometimes sometimes not you got to work with it so you create this and you go hey let's make it zero times infinity no let's make it Infinity over 1 time Z that's infinity infinity over infinity and then loms okay we got to we got to get on with this stuff uh we're going to talk about continuity right now listen here's the rundown on continuity um if function is continuous on a region if at each point in the region we can find this this little neighborhood around it where we don't have any gaps jumps or boundaries so basically we say if we have this point on the interior of a you don't have to draw this I'm just going to explain it real quick if we have a point on the interior of a region you can always draw a little neighborhood neighborhood Delta a little that just means a little neighborhood okay just a little circle around it and even if you got really super teeny cloes you could still draw a little teeny circle around that does that make sense to you you can Define some points around it which means that at that point that we can draw this neighborhood around we can get to that point no matter what path that means that we're continuous on that surface over that region on Boundary points so if you're actually on there go well this looks pretty good but these points out here don't look so hot in that case we say you know what we're going to restrict our paths just restrict our paths to all paths that come from the inside sorry inside the neighborhood like this we say paths outside don't worry about those paths outside so boundary points H boundary points are okay as long as we have a strict our paths all this basically says I know I'm going speeding through it because it it matters it does matter but it's easier than than some of these definitions make it here's what this says if you have a region you have a function that's defined over this region it's going to be continuous at any point in that region why because you can draw a little circle around it that's that's continuous within that Circle that's why on the boundary you're still fine because you can find some paths from inside the region that get there in other words all of this stuff says says the following a function is continuous at any point inside of its domain that's all it says here's a domain here's a point continuous throw a little neighborhood around there all the paths go to that point no problem well to get to that point uh by any path it's going to it's going to it's going to be there there's no jumps there's no gaps there's no asmos there's nothing wrong in that if we're completely defin that domain same thing on the boundary there's no jumps or gaps with this part of it we can find continuity there all it says is that a function is continuous at any point inside the domain that's what I want you to write down right now so as far as continuity goes a function is continuous at any point inside the region for which it's defined um domain for well how do we Define the regions how do we do that on what regions are functions defined we already did that it's called The Domain find the domain you find out where the functions continuous do you remember that back from Cal 1 when you're like wait a minute finding continuity issues remember holes and ASM tootes from Cal 1 you watch them on the videos you probably you probably saw that but where we found discontinuities were exactly where we found domain issues they were basically one and the same until we start restricting our domain if we don't restrict the domain domain and continuity give you the same region and that's what I'm saying here saying that a function is continuous at any point on the region for which is defined the region for which a function is defined is defined to be the domain this is this domain and and that's kind of beautiful um because we get a lot of the continuity properties we had before they still work all we have to look for are the same exact things we looked for before holes these weird things that would be like denominators equal to zero uh negatives inside square roots so just look for the same problems we looked for for domain does that make sense to y'all okay uh there's a couple things I want you to to write down here couple things that we get to use the first one polom are continuous everywhere rational functions are continuous everywhere the denominator is not zero and continuity holds for compositions of continuous functions so just put continuity holds for compositions so those are the three things that I want you to put pols continuous everywhere rational functions are continuous everywhere where the denominator is not zero pols continuous everywhere rational functions continuous everywhere the denominator is not zero and three continuity holds for compositions of continuous functions this last statement is why I can't do the squeeze theorem until like right now we're going to do some some continuity and then we'll talk about squeeze theorem because this continuity holds for compositions of continuous functions I need that to prove it uh so we're going to we're going to do that in a little bit right now uh I'd like to do one example just one okay by one I mean two uh two examples about continuity then we'll take a break we'll come back maybe do one more we'll do that squeeze theum thing I'll talk about compositions and uh and that'll be it pretty close I do want some more energy out of you guys though you're making me feel like I'm punishing you oh crap you do feel like I'm punishing you question where would where would domain issues and so likewise where would continu ISS continuity issues exist on this problem if anywhere where would they be good polom continuous everywhere perfect polinomial continuous everywhere rational function continuous everywhere unless the denominator can equal zero now here's my question can the denominator equal zero not without imaginary numbers do we have imaginary numbers thank goodness no uh so we don't even have those so are there any continuity issues so this would be continuous on let just put it this way on All Points continuous on all ordered pairs XY question F XY oh yeah I'm sorry thanks for that hey if you're okay with that one how about this what if uh what if I just do one thing is that still continuous so notice the change please is that still continuous on all ordered pairs no what's the one point that I couldn't have 0 0 so this would be continuous and here's how you'd write that write continuous on any ordered pair such that that ordered pair does not equal the point 0 that's how we'd show the continuity say hey anything Works only one that doesn't work is when the denominator is zero that's a polinomial can no problem if it's zero who cares but this one can't equal zero that only exists at one spot that only exists when XY equals the exact point0 Z that's my only discontinuity should fans be okay with with that one okay here's the second example when we take it right please notice the compositions please notice that they're really not an issue um as far as creating for you more discontinuities all they do is they might have some within so let's let's take a look at it what do we know about this this thing right now firstly is e to the x continuous everywhere e x is a function that's continuous everywhere for sure is X over y continuous everywhere what's the problem here what couldn't we have come on quickly what could so we can't let y equal Z so y cannot equal Zer tell me right side something else quickly what else do you know about this thing could it be zero so that that's that's it you that's literally it the only things we have here are X got to be greater than or equal z y cannot be zero so we have any order pair such that X must be positive and Y cannot equal zero sense domain basically sense domain gives you the the region where your function is defined and since the region where your function is defined is what's give you continuity basically finding your domain is practically the same as finding your continuity in almost every single case and that's that's what we do with we'll do maybe one more when we come back from a break but I think we uh I think we need a break here real quick show pans feel okay with what we're doing so far okay so a few examples uh we're talking about continuity where we're finding out is that finding continuity and finding the domain most of the time are pretty much one the same huh almost Rhymes I like that so um your just looking for the problem issues that's all we're doing now that is holds true for functions in three independent variables so let's check out this domain here's the problem check out the continuity here's the issue just like domain of two variables is on a plane 2D domain of one variable is on a line 1D domain of three independent variables is 3D which means continuity is also a 3D issue so let's take take a look at this do we have any domain issues on the numerator yes no denominator yes no what do you know about denominators everybody come on quickly so I know that that's true that's it that's the only continuity issue we have so write it down we want all sets of order triples such that x^2 + y^2 + z^2 cannot equal four if you add four to both sides heading off if you're okay with that one now think about what this means think about what this means can you tell me what shape I'm going ask you what shape this isn't um what shape is X2 + y^2 + z^2 equal 4 what shape is that it's a what id what type of ellipsoid since all of our numbers are exactly the sameere it's a sphere here's what this says says any point in all of three space will work we have no discontinuities there except for this bubble of that sphere all it says is that the points you can't plug in all those points are on that sphere does that make sense to you it's all of the 3D space the the discontinuity exists on the on that sphere uh the points that create discontinuities are on that sphere that's why we can't let it equal a radius of two so F feel okay with that that's it that's all we're doing it's just regular continuity now what but I do want to tell you the the statement three down here a little while ago continuity still holds for compositions and parametrics are all compositions so when we're going to try to figure out some continuity that has compositions in it just remember one thing you can't ever make domain or continuity issues better you can only make them worse that's encouraging right you can't fix a continuity issue ever um here's how to do some parametrics some compositions what ever you want to call them let's focus on finding h of XY such that this equals g of f of XY in other words what I want to do is do you know how compositions work this says uh take the F thing put it in for the G thing so take this entire thing and replace this variable and this and this with it do you actually want to do that no not really all I'm worried about is what's the continuity going to be for this composed function that we're going to get here's how to do that firstly find the continuity here find the continuity here and I'll show you how to work with it what's the continuity great why well all all I I don't say all real numbers all order pairs that's a better way to say it because these are order pairs so this is continuous on all ordered pairs true or false this is continuous on all T yes no no okay what's it not continuous at say what plug in plug in anything you want can you do it can you plug in anything to cosine can you plug in anything to sign can you plug in anything to T no issues oops continuous on all T here's what I want to tell you if this please watch carefully what we're going to do you're not watching you should be watching please watch carefully we're going to be taking this function and plugging it into this function does that make sense everything would be ending with x's and y's the t's would disappear that's what this says replace T's with XY stuff you get me so lots of XY if this is continuous everywhere and this is continuous everywhere tell me where this is continuous everywhere but we're going to be in turn terms of whatever that says so h of XY that's why it says XY because you're going to be ending with XY this would be continuous also on all order pairs does that make sense with to you that track with you now let's try this one it's a little bit more complicated uh the first part's really easy true or false this is continuous on all X Y yes no yes true or false this is continuous on all T yes no continuous on t such that what such that t is let's make it really simple this says greater than or equal to zero this just takes away the equal so be okay with with that one now here's what's going on when we find the continuity for H of XY such that this equals g of f of XY what this says again take your F function and plug it into your T function here's what that means it means every place you have t what are you going to have you're not excited should be excited every place you got T you're going to have what no you have X Y you're going to have an entire function what entire function this says for every variable in this function you're going to plug in this entire freaking function there and there does that make sense in other words you're going to let t = x - 2 y + 3 does that make sense to you now here's the magic part about that what do you know about t read it what do you know about T it's got to be greater than zero that means that t must be greater than zero therefore this thing must be greater than zero can you can if you can see that substitution then you got it made can you see how to get from here to here T's going to equal this right because we're going to take that function plugged in for T if T is got to be greater than zero by what we have if T is got to be greater than Z and T equals this thing and this thing's got to be greater than equal to zero why are we doing that in terms of X Y because the function is going to end in terms of X and Y does that make sense now solve it do something just subtract the three or something so we got x - 2 Y is greater yeah flipped it around x - 2 Y is greater than -3 does that make sense so we found continuity this will be continuous on all ordered pairs X Y such that x - 2 Y is greater than3 so what we do is we figure out continuity if it's all real numbers great all order pairs I should say figure out continuity here if it's not just set the parameter equal to what you're plugging in for that parameter and then you have this this anality solve for the inequality and you have the do the domain and the continuity for that that second composed function heading enough for okay with that one now last one this is going to be the opposite case let's start here is this continuous on all T yes or no is this continuous everywhere no notice it's tangent is it continuous on all X continuous on all y what's wrong with it ASM tootes where all that values of Pi 2 so we have this is continuous on ordered Paris XY such that X I don't care about X is good but I just can't have y equaling < / 2 I can't have yal pi/ 2 or any multiple of that so we're going to say plus K pi and what that says is if you add or sub ract pi to it you still have a problem any multiple or k is integer Pi you go okay so pi/ 2 or 3 Pi 2 or any Circle any like half circle around there you can't have any of those now here's a deal here's a deal watch carefully if I want to talk about this here's what I want you to see this is going to be in terms of XY correct is this going to add any more continuity issues to this situation so this this added okay this said I have continuity issu I didn't so we have to set equal to this continuity issue have to figure it out if I say I'm taking this whole thing and I'm just going to substitute this in for T this doesn't add any more problems all it says is that yeah you're going to have you're going to have continuity basically the same region you had this this was the only problem there it was already in terms of x's and y's so it's going to be the exact same value of points does that make sense to you the the cosine is continuous everywhere it's not making a difference for us it's this piece that even when I put it here I still can't have the Y equaling those multiples of pi/ 2 do you guys catch that so composing you basically keep the same continuity sometimes you have to work with it and say okay well my T's are changing changing to X and Y figure that out but for the most part need to take continuity from both of them and kind of jam them together that's basically what we're doing so this says continuous everywhere that means I'm not going to have any issues when I plug into this function I'm not going to have any issues when I plug into this function we look at this function okay this is what I'm plugging into that where's this not continuous well it's continuous everywhere except this therefore since this doesn't create any more issues this is also continuous everywhere except that should F be okay with with that one okay I promised you this I promised that we were going to talk about the squeeze there I'm show you one example I said you know when we uh when we started this so it's really really hard to prove that limits exist it's really easy to prove they don't exist we spent most of the time doing that how we prove limits exist is by using continuity and just plugging in the number that uses continuity we talked about that a little bit we said if a function is defined then it's got continuity if it's got continuity the limit exists so just plug the number in and if the function is defined there then it's continuous there we talked about the regen we talked about how if we're defined and and we plug that point in over the defined region that we find a neighborhood of points and we're continuous there the limit has to exist so if we're to find continuity no problem just plug the number and you got it well what if if we can't do that and what if the limit still exists and we didn't have the magic trick of polar this is the case that you use so if we go if we don't have that stuff for example let's talk about this limit only one first thing you should always do with limits is what PL plug it in and you plug it in you get zero over zero you can verify that form if you want to and then we go okay we're used to saying things don't exist in this case but this one does how do you prove it you might try polar and see what that gives you you could try it that that might work I don't know I want to show you in this case how to use a squeeze theem to your advantage this is the other way so the things we do with limits first plug it in if you can plug in and you get a number then the the functions defined there which means we're continuous there we just plug it in we're good because we're continuous limit exists got it um that's fantastic or we try to prove it doesn't exist by finding those different paths we spent most of our time doing that or using the polar trick that we we created that was pretty cool or or the squeeze theem those are basically the things we do with limits I'm going to show you the squeeze theem right now so let's take a look at this let's let's look at our function and I want to can I want to consider the absolute value of the function like where are you getting that from I'll show you where I'll show you where in a second true or false this denominator is always positive no matter what the absolute value doesn't really matter true or false 5x^2 is always positive no matter what that doesn't really matter true or false Y is positive no matter what false so this would equal 5x^2 over x^2 + y^2 absolute value of y does that make sense to you now we're we're going to build some stuff here I don't want to confuse you but we're we're going to build from scratch and try to get back to this thing so we're try to get here so here we go this is going to seem kind of nonsense but you understand this right that this is equal to five 5 X2 + x^2 is equal to 5 you go okay why why are we doing that well that implies this if I add anything to the denominator if this equal 5 and I add something that's greater than or equal to Z here is it still equal to 5 what would it be smaller than five smaller than or if it's zero it would be equal to five does that make sense true or false y s is greater than or equal to Z so since y^2 is greater than equal to Z if this equals 5 and I add something to the denominator I make the denominator bigger I make the fraction smaller it's got to be less than five or at most equal to 5 head now if you're okay with with that one okay we're going to use that in a second now this is why we have to have the absolute value what am I you know I can do anything I want to with inequalities as long as I do it to both sides and If I multiply by a negative I've got to flip them right do you know that okay I'm going to multiply by y now that's a problem it's a problem if I do that watch the problem if I multiply by just times this by Y and times this by y that's a problem because y could be positive y could be negative I wouldn't know I wouldn't have a correct inequality that would be a big big situation does that make sense however if I force this thing to be absolute value now now I'm multiplying by what type of number for sure positive do I have to flippy dippy the inequalities no no okay what it gives us is 5x^2 absolute value y oh my gosh over x^2 + y^2 is less than or equal to 5 absolute value y can I have an honest show of hands you feel okay with with this so far this is the reason why we needed the absolute value was so that I could multiply by something and not flip that inequality and get the same thing out of it you guys get the idea that's why I needed that now let's let's think about this tell me something we're going to start with this tell me something about tell me one thing about that that you know it's on the board already tell me one thing about that that you know always positive it is always positive so it's greater than zero tell me the other thing that we just found out say that again how the squeeze theem works is this it says if you take the limit of this and you take the limit of this and they go to the same number the thing squeeze or sandwich theem something we call it the thing sandwiched or squeezed between them has to go to the same value what's the limit of zero as X Y approaches I don't even care come on it's not your question what is it and what's the limit of five absolute value of y as XY approaches 0 0 notice there's no X but I don't even care what's it go to by The Squeeze ther sance the limit of this function goes to zero and the limit of this function goes to zero the limit of this function Z by the squeeze them zero zero we built it zero zero midal's got to be zero now we haven't proved a darn thing right now but we're about to you guys okay with this you sure no yes I want you to look back at this piece this piece is equal to that piece do you guys follow that yeah know therefore the limit of the absolute value of all this jump is zero and you awesome but wait Leonard wait the leard is this exactly the same as that no no but this is why we had statement number three over here like 25 minutes ago it was this continuity holds for compositions of continuous functions here's my question to you is the absolute value function a continuous function do you see that this is a composition of this crappy function inside the absolute value function you guys see that continuity holds that means we can do this stuff with it but the big Point here says that means that this limit equals the absolute value of the limit of 5x^2 y X2 + y^2 x y 0 0 awesome that's the same thing absolute value of Z is zero we get the same exact thing so we have this composition of continuous functions we prove that limit exists by The Squeeze theorem I got to stop it right here and and we're going to call it good on this section