so let's get started with our our actual calculus stuff oh this is going to be awesome so with our calculus we're going to talk about limits first now our limits is really the basis to our calculus it's how we do calculus we're going to find out some kind of tricks to calculus later but for right now if you know how to do limits you're going to be able to do calculus so it's probably a pretty important idea yeah yeah so we're going to find that out towards the last part of this class uh for right now I want to lead you through in introduction to calculus and what it's all about are you ready to learn that so r view is over with thankfully now we're going to slow down just a little bit uh we're going to talk about actual calcul stuff so no more really hair raising kind of oh my gosh so much stuff uh we're going to go a little bit slower but it's going to be more in depth you ready for it slightly slower not not much slower in calculus we have two basic goals in calculus one this this introductory course calculus so here's our two goals goal number one the first goal and the one that we're going to spend most of our class on at least the first half of our class on is this school given any curve given any curve not straight line straight lines are easy but given a curve I want to be able to find the slope of a curve at a point isn't that the interesting idea if I say to you I want you to go ahead and find me the slope at that point right now we really have no idea how to do it I mean you can approximate it you can go oh okay I'll find your slope let's see slope is2 is that accurate no idea right we can't really just kind of approximate slope what we're going to do is find out a really good way using calculus on how to find the slope of a curve at a point in fact when I ask you a certain thing in this class you're going to say oh it's the slope of a curve at a point we're going to talk about that for a long time that's goal number one goal number one is to find the slope of a t or to find the tangent to a curve at a point the tangent is the line that intersects a curve at one and only one spot in a in an area so goal number one find the tangent which involves the slope of the curve at a point that's goal one if we can do that we're we're on calculus fast pass Fast Pace that's that's good so we want to take a curve be able to find the slope of that curve at a point that's going to lead you the tangent and and vice versa tangent and slope uh go hand in hand for for us goal number two which is the last half of our class maybe a little less than that goal number two is it's also a really interesting question so cool by the way this stuff is stuff you can't answer in any of class right you can't you can't do that with algebra there's no way you can do that with algebra bro you can't find the slope of a curve are you kidding me slope of a straight line easy slope of a curve wow it's weird that's is that not an intriguing idea to you it should be if you're in this class like that's kind of it's kind of cool how you going to find SL of a curve I don't know calculus the next question is probably even a little bit more interesting next question is let's suppose I have some funky any curve that that you can think of as long is defined by a function can you between two points find the area under that curve can you do it the answer is right now with what you know can you do that with geometry no because I could say okay great uh find the area of something that's curving can can you do that no no you can't do use the area for a rectangle it's not a rectangle you can't use the area for triangle that's not a triangle you can't break this up in any way that you have geometric figures can you not even circles not even with radiuses because the radius is changing so the next question is can you find the area under a curve those are the two goals of calculus find the slope or the tangent of a curve at a point and secondly can you find the area under a curve between two points interesting stuff weird stuff right it's crazy questions going on here and amazingly it's actually not that hard uh this is I think it's easier than uh pre-calculus it's easier than that you don't have to do maybe not easier but if you know pre-calc this is not such reach what we're going to talk about first we're going to talk about the tangent problem I'll give a very intro to this but I'll talk about later on we're going to talk about this for most of our day today so let's talk about the tangent problem are you ready do you have any questions on the ideas of the goals of calculus do you understand what the goals are the goals are a find the tangent or the slope of a curve at a point that's what we want to do goal number two after we finish this we're going to move on to finding the area under a curve between two points very cool ideas so let's work on the tangent problem right that you you're lucky you're here today okay this is this this is like the fundamental idea of calculus this so cool we're going to figure out what a limit is from doing this idea right here too here's the tangent problem the tangent problem says if you give me any arbitrary curve so I'm just making up I don't even know what that that is it's just arbitrary curve and you give me a Point let's call that P give me a point call it P I want to be able to find the slope of that curve at Point P did you get the idea talking about the tangent problem right now right slope of that curve at that point because if we have the slope and we have the point we can make the equation of a tangent do you get me that's that's pretty easy that's just y - y1 = MX X1 so right now we have the point what we're worried about is the slope of the curve at that point do you follow now the problem is this um if we're going to make up a tangent line how many points do you need to make up a line how many points do we have that's the problem right how are you supposed to make the equation of a line when you only have one point and you don't know the slope yet do you see the issue that's the problem so here's what we're going to do we're going to say okay okay well let's let's put some point somewhere else on this thing how about uh Q up there if I connect these two things I connect those two things do you remember what what is called when you connect two points together it's not a tangent it's a genome it's a line if you've had some geometry a secant connects two parts of a well usually a circle but of any function so we would say that PQ is a secant line it touches two points of this curve now here's the question right you you agree that we need two points for a for a a line yes but we also agree that we want the tangent of Point P that's the ultimate goal here so my question to you is is PQ a good approximation for the tangent is this a good approximation for this okay it's an approximation right they're both positive okay sure what I'm asking you is is there way think about this for a second is there a way that I can make this secant approximation better to make it more approximate the tangent can I make this a better approximation how could I do it let's assume for a second that P is fixed but Q is movable you can move the point Q I can't move P but I can move the point Q can you move Q to make this seen line a better approximation to our tangent line can you think of that where would you move Q would you move it up would you move it down let's look what happens this is going to be interesting so we're going to find out what happens as we move Q down this line so if I move Q here is that better or worse better because I'm going to have use different colors here that this would be like Q sub one maybe a different point for Q uh if I move it closer is it going to get Better or Worse better okay if I move it closer it's going to get even better if I move it closer it's going to get even better if I move it closer it's going to get even better when do you stop can you move it really really close can you move it so q and P are the same point why not cuz then you only have one point how many points you need for line we need two so can we move it close answer is sure we're we're doing it right now can you move it close yes can you move so close that there's no difference between them yeah can you make it the same point no that would fail because we need two points I don't care how close they are but you need two different points right if you have two different points you can make the equation of a line that's great that'd be fantastic that's our idea actually so we're going to try to move the point Q really really really really really close to the point P if we can move it really really really really really close that's math speak by the way really really really really means very very very very if we can move really really really close then our secant line is going to be really really really really close to a tangent line do you agree with the idea so you take this point you go let's move a really close not really far away that's a sucky approximation but if we move it really close move that secant really close that point really close that secant is so close to a tangent line that it's not going to make a difference that's the idea so let me write some of the stuff out for you so what we're doing right here we need two points for a line sure we understand that awesome so we have the points p and Q well the question we're asking is what happens as Q gets closer to p and the answer is as Q gets closer to P but we just talked about this we just said this I'm nothing new as Q gets closer to P the secant line more closely approximates the tangent line now have you understood that at the first okay so as Q approaches I'm going to use that word we're going to use that word limits as well approaches that means gets closer to as qo approaches P we get closer to a tangent or an in other words the secant more closely approximates the tangent okay well that's great that's great oh one more little note why was it that we can't just let Q equal p what what was the reason why can't we just say oh let's move it to P because there wouldn't be a difference okay there there wouldn't be two actual points to be the same point twice and that's that's a problem you couldn't make an equation of a line with that so one little note we can't just let Q equal p because we need two points to make a line you need two points that would be a problem here's the idea of a limit all right this is a big big idea here's the big big idea for limits okay the idea is how close can you get one point to another without them being the same point so let's say I have two points like this can you find some space between it let's say I move it over to the midpoint can you find some space between it say move it over to the midpoint can you find some space between it can you keep doing that for eternity between any two points there's there's a gap in that Gap you can fill it in with another point because points have no bread that means that even though we draw them on the board like this they really don't have any distance between them so two points can get would you agree infinitesimally close to each other that means so so close that you can't put anything between them but there're still some little distance and then you can get even closer that and closer that and closer that would you agree that to that so here's the idea behind calculus it says we can get Q so close to P that there's literally no difference between the secant and the tangent we can get it so close we can't let it equal but we can get it so close that the secant line is the same as the tangent line that's the idea of a limit isn't that kind of an interesting idea yes they're different points but they're so close doesn't matter that's the whole idea so the the big big picture for our limits is if we let Q get really really really close to P the secet will be identical to a tangent that's the idea or approximate it so closely that it doesn't matter really really really if we let Q get really really really close to P I'll use a better word I'll use a more Mack word for that just a minute really really close to P the seant will be identical to the tangent here's the idea right here here's here's the idea now replace this really really really with with a key phrase for you the idea of really really really the idea of moving P or Q really really really close to P is called a limit moving something really really really close without actually getting there that's the idea of a limit you kind of understand the idea of a limit we would say that Q would be in a limiting position that means we we've got it all the way really as close as we can to P but it never touches P that's the idea of a limit it gets really really close so the idea of really really close is a limit in the most simple terms this is it would you like to see an example of how to actually do this would that be interesting to you how to find the so Curative point you want to see that it's kind of fun okay we're going to work through this um I'm I'm simply doing this to give you an understanding of how limits work and the jump that you can make okay uh we're not going to be doing a whole lot of math like this I'm just giving you an introduction right now so that you see it's possible with some real math stuff that you've had before without actually teaching you limits and calculus I'm going to make one limit jump in this problem I'll show you when it is uh but this is stuff that you could actually do we're just going to we're just going to use calculus to do it better you ready for it do it more mathematically before I go any further did you guys understand the tangent problem not your if you're okay with the tangent problem you understand the idea of a limit moving a point really close to but never actually getting there again why can't we get there need two points yeah that be okay cool now we're ready I'm excited first little step off into back this land do you trust me you shouldn't here's the goal to this problem I want us to find the equation of the tangent line to this curve at a certain point specifically that one one one well let's go over to 1 one and let's put a point there so p is the .11 I'm going to walk you through how we're going to do this right now okay I'm going to walk you through how we're going to make up a tangent line using this idea are you ready for it over here on this idea we had two points right P set in stone you're not going to move P what other point do we have we're going to make a point q q what's the coordinates for Point Q tell me it's coordinates for Point Q we can't use actual numbers right that would that would be very good so I I I know 24 would be on there and 39 would be on there 416 and so on but I want you use actual points in general what are the points for what are the coordinates for any point XY very good okay XY so now write down XY and have your eraser handy have your eraser handy because I'm not going to have X Y I want to keep this in terms of one variable Now using your your your knowledge of what the function is what's the function y equals what wa say it again y equals what y equal so instead of having X comma y would it be okay with you if I had X comma X2 yes no cuz Y is X2 right so that would't matter so erase that and I want you put X squ cool all right all right now we have this secant line going from P to q p is a fixed Point 11 one Q is a movable point x comma X2 would you agree that any point on this line will have the coordinates X comma X2 no matter what I plug in right so that that movable Point that's going to have the same coordinates all the way down now we have this secant line here's what we want to know we know that the equation for a line is y - y1 = m x - X1 true what we're trying to find is the equation for a tangent line here's what the equation for a tangent line would be pay close attention it would be y - y1 that's the same equals the slope but a specific slope it would be the slope of the tangent line true what if the tangent line slope happens to be so the slope of the tangent line and then x - X1 now the cool part about this I I'll I'll recap this in just a bit the cool part about this is we already have a point what's my fixed point that I have one one I already have one one so really what this comes down to is can you find the slope of the tangent that's why I said finding the slope of a curve at a point is the big deal for calculus you already have the point that's the easy part the slope is the hard part so we're going to try to find right now the slope of the tangent are you ready for it m yes you sure we're going to use the idea of SL of a secant so we're going to take a break right now we're going to come back to this uh in case you're just a little bit lost here's what we've done so far this is y Square I gave it to you we fixed one point according to my problem that's 1 one we've made up another point a movable Point q q is that X comma x s where did the X squ come from that's cuz it's y all right great we now know the equation of a line is this so therefore the equation of a tangent line would be the same exact equation it's just we'd have the slope of the tangent line the problem is it's very hard to find the slope of a tangent line without doing the limiting idea you can't do it because there's only one point there so we're going to have to use the idea of a secant and make it into a tangent line by moving Q really close to P do you get the idea basically doing that with this problem so let's take a look at the slope of a secant our specific secant hey by the way do you remember the slope in general the slope formula slope is what over what okay great in terms of our coordinates slope is Delta y Delta X specifically in terms of our coordinates what is our slope that's what I'm looking for Y 2 y1 yeah over X2 - X1 otherwise you get the the negatives of that slope so Y2 - y1 X2 - X1 do you follow how people feel okay with this so far just slope formula what we're going to do we're now going to find the slope of our secant we've got the secant on the board right the secant will be the slope I'm sorry the slope of the secant will be the slope from points P to Q agree let's plug in these coordinates we only have on the board plug in these coordinates into that formula in our case our Y2 is what our Y2 would be x s what's our y1 not X be one what's our X2 what's our X1 okay Y2 - y1 over X2 - X1 not if you're okay with that so what we'd have for the slope of our secant is Y2 - y1 over X2 - X1 are you okay with this so far find out where the slope comes from there's a couple notes I want to make at this point first I'm reiterating a lot of stuff here I know that I am I'm doing on purpose CU these are the key points of calculus all right you need to understand what we're doing I would hate for you to get through this C class right and at the end of it know how to take derivatives and integrals and have no idea what you're doing you can be successful on just doing derivatives and integrals in this classroom but if you don't understand what it is you're actually doing with those things you don't even know what they are right now but if you don't understand what you're doing it's irrelevant you're just now just doing formulas it sucks if you know what it comes from what you're actually doing I I'm going to make sure you know by the way it's a little bit more interesting and you can apply it to more things do you get the picture so right here what we're doing step number one is if we move Q close to P this is going to become this do you agree as P I'm sorry as Q I'm going to use a new symbol here approaches P that means as Q gets really close to P without touching the slope of the secant line approaches the slope of the tangent line would you agree with that statement that's over here right now you you agreed with it already haha tricking uh as Q gets close to P the secant gets close to the tangent true that's that's what this says right now okay as Q gets close to P the secant gets close to the tangent how close it depends on how close we get these things if we get them so close it doesn't matter then these become so close it doesn't matter agreed it's kind of neat right it's a cool idea now the the problem is here's here's note number one here's note number two this is interesting right where are we trying to find the slope what point one 11 one well and what's the x coordinate for one one to find the slope of anywhere you you plug in that value right this gives you the slope of the secant line true plug in one plug in one what happens no you don't get zero you get zero over 0 over zero is not zero that's a big problem 0 over Z 0 over 0 is undefined right isn't that an issue this is why folks this is the reason why why you cannot have the point getting listen you can't get Q close to P because if you plug in one it's undefined that means that you're you you have the same point so you can't let X = 1 that's big time that means double important you can't let x equal one this is why we can't let Q equal p because we only have one point right you can't find the slope it fails it's undefined at that point this is why Q can will be if it does if you try to find SL of the SE at that point move Q all the way down and then plug in the x coordinate if it's the same thing you're going to get something that's undefined because you're trying to find the difference between points that don't exist a difference that doesn't exist and that's going to be you're divided by zero the there's no difference on the x axis that would mean you're dividing by zero you get something that's undefined you get an undefined slope that's a bad thing so we can't let Q get all the way to P true all right what can we do now this is going to blow your mind you ready to get your minds blowing like a mind grenade you ever seen Yes Man watch yes man so funny had a rib bull you have a rib bull let's go rle have you not seen it that just sounded really stupid you watch yes man it's pretty funny now so next part going to blow your mind can you factor that in fact learning what we knew about ASM stuff like that you know that if you have 0 over Z it's factorable because of well some stuff in mathematics that I haven't bored you with yet but if you ever have uh the some number that that makes zero it means that that's a factor if you have some number the same number that makes two polinomial zero it means they have a common factor we've got a common factor here in fact if you were to factor this this is going to give you x + 1 x -1 over x - one agreed okay do you think see anything that simplifies out of that aha now this is cool now this this is interesting very interesting what we're going to do is we're going to be able to simplify out this thing now I know what you're thinking well wait a second can't you not simplify out a domain issue isn't that a problem for you and the answer is yes no it would be a problem normally however watch carefully are we actually letting equal 1 is Q actually getting to the point p no so is X actually equaling one no we're getting really really close I'm talking like 1.000000 forever and then a little bitty one at the very end of it really close but it's never actually equaling one so keep in mind when we do this we're not getting rid of any do we're not really altering the domain whatsoever because we already knew X wasn't equal 1 so we have this little restraint already X isn't going to equal one so then what we know is okay the SL of the secant now equals x + 1 were there any questions on this because I got to eras it do you have any anything all right is this making sense to you do you see where the slope formula came from do you see how we can factor and simplify it and now we get down to here we don't have any problems we not eliminating domain issue because we're actually not letting X get equal to one it's just getting really close to it here's the jump that we're going to make okay here's the jump so that was true that's what we have down what I'm asking you is as Q gets closer to P so basically as X gets closer to one uh what happens as Q is closer to P that means the X variable is getting closer to one as X gets closer to one can you tell me what happens to the value of the secant let's try some okay think about this for a second just just do a little bit of math with me let's say we started at the point uh 416 so X would now be four plug in four here how much would you get okay now let's move it down to three plug in three here what would you get move it down to two how much would you get okay move it down to 1.5 how much did you get 2.5 move it down to 1.3 what would you get move it down to 1.1 what would you get move it down to 1.01 what would you get. 2.01 good move down to 1.001 what would you get move down to 1. Z forever and then a little one at the end what would you get 2.0 for every little one wouldn't you would you say that as this thing gets closer to one the slope of our secant gets closer to two would you say that because I'm plugging in things really close to one I'm going to get out things that are plus one really close to two does that does that make sense to you our sequence going be really close to two here's what that says this lets us make the jump this is a limiting position it says that we know the limit of the slope of the seant line is two what that means for us is that the slope of the tangent line actually is two that's the jump it says the secant line at this point if x gets super super close really really close if x gets really really close to one the seant line gets really really close to two now I can't let x equal 1 but I can make the jump that if I could let x equal to one the slope would be two that's the jump there do you guys see the jump this is the the using limits to make the jump between a secant and a t tangent so because the secant's approaching two we say okay we got this that's called a limit we're going to talk more about limits later on but the jump is going from here to here now can you fill out this equation using that information sure we know that it would be y - y1 which is 1 equals the slope of the tangent hey we know it's 2 X - 1 now of course we don't generally leave things in point slope will solve them do you see where the one the one and the two are coming from folks M our point is 1 one our slope is now two that's the slope of the tangent if we solve it we get y - 1 = 2x - 2 add the one y = 2x - 1 that's the slope of a curve at a freaking Point that's awesome oh my gosh Isn't that cool we found the first tangent line to a curve that you've done now granted it's not a very hard curve but here's what it is I I'll show it to you you can actually graph it here's minus one it crosses there you go up 2 one two over one oh that's the point and if I graph it it intersects at only one spot that's it that's a tangent line isn't that awesome your mind should be blowing is that why you're looking at me stunned like yeah is that your stunned face it should be your stunned face okay now by show hands after 40 minutes of doing one example how many people are show we talked about good you have down right now now the basic idea for calculus this is the basic idea using limits to find this tangent of a curve at a point is it going to get more advanced in this yes of course it is but that's the basic idea and we're going to do things differently when we actually get to the calculus we study limits more first but we we will get there now the the next thing we got to talk about is the different problem I'm going to I'm not going to spend a whole lot of time on I'm basically just going to introduce it to you cuz I I know for sure you guys are not going to remember this uh by the time we actually get to areaid curve so I'm going to to re introduce it to you again but for right now I want you to just get the idea about what's going to happen later on okay so the area problem here's the area problem a problem says can you find the area under a curve between two points the answer is yes you can with Calculus only how are we supposed to do this thing and let limits work for us here here's the the plan uh firstly can you find the area the way it is right now no it's got a curve to it you can't find areas with curves in them so so what maybe would be an idea you can approximate it by making it like a rectangle oh I could yeah sure if I did this and said okay from here I just want to make a rectangle would that be an approximation mhm would it be a good approximation no not really however what if I did okay I don't want just one rectangle maybe I do this I say okay I want a rectangle from here to here then a rectangle from here to here then one from here to here here and went from here to here is that a better approximation the missing area is smaller yes that's the idea behind the area problem what we're going to be doing is making rectangles why rectangles well let me ask you can you find the area of a rectangle that's why rectangles and they're easy to draw we like that so if we make lots and lots and lots of little itty teeny bitty weeny little rectangles like that all the same width but going the entire length of our our curve and we add up all those rectangles are we going to have a pretty good approximation of the area and in fact if we make those rectangles infinitesimally small so whereas you couldn't even slip a piece of paper between them and then add them up is that going to be even better in fact if we stuck an infinite number of rectangles between this point and this point which you can do with limits we're going to have a perfect area and that's the idea for the area problem you stick an infinite number of rectangles in there and add them up it's going to involve limits because we're letting some number go to Infinity you can't ever reach infinity but a limit will take care of that for you that's the idea behind the area problem does it make sense kind of the idea of the rectangles trust me we'll get much more involved later you have no idea how to do this right now don't worry about it okay we're not even to do a problem we'll do that later chapter four or something like that so let's define a limit uh what a limit says in English is what does the function do as a variable approaches a given value that's what it says in English so limits what does the function do as the variable approaches a given value that's the question now do we care what happens to the function at that value the answer is no the limit isn't about getting to the actual point it's about what happens as you're approaching that point getting really really close you see the difference there in the previous example we couldn't actually get to one do you remember why some of you're zoning you're zoning out can't have one point on the line we'd also make a a undefined point right it be undefined that would be bad but we we saw what happens when we get really close the function got really close to something else that's the idea of a limit so we don't care what happens as you get to the value what happens as you approach the value um for us this is this is exactly what we did actually kind of in our heads what happens what happens to X2 as X approaches let's do two that's X approaches two I could written in English but that's just as valid what happens as X approaches two now do we care what happens if you plug in two no don't care what I care about is what's going to happen if I draw this table right here with X and F ofx and I say I'm trying to get to two I don't care what that value is what I'm trying to see is what the function is doing as I'm getting close to that thing so what's happening as we approach it from the right what's happening as you approach it from the left give me oh I already ruined it give me some numbers uh to the right of two two numbers to the right three give me something smaller than three like 2.5 is I like that that's what I was about to write almost ruined it give me something a little bit closer to two closer than that maybe 2.1 yeah or 2.01 we want really close right so 2 .1 and it maybe 2.1 that's pretty close take your calculator out by the way now let's go the other way give me a number that's smaller than two that we want to work our way up from this way okay sure how about we we keep it kind of symmetrical 1.5 I like that one we'll do uh 1.9 all right and 1.999 how about that do that work for you take those numbers our function's x s let's look what happens to the function as we plug those things in if you plug in 2.5 what do you get when you 6.25 or something like that plug in 2.5 you get 6.25 yeah okay so we'd start with 6.25 what we care about is does this side look at the look at the board if we go this way and we go this way do they meet up at the same point that's what we're talking about for limits now this is a very very easy example we just were working with X2 we know the answer is going to be four right Le it should be four we're going to see if that actually happens when we when we evaluate our limits plug in 2.1 someone out there tell me what you get when you plug in 2.1 what you 4.41 okay that's Clos to four tell me what you get when you plug in 2.1 2. wait what is it 4.41 how that like that how about 1.5 oh that's that's 225 isn't it how about 1.9 plug in 1.9 what do you get when you do that X S6 36 and now do 1.999 1.999 what do you get how much 3.96 34960 one M yeah okay cool what's going on here let's see with the limit we really don't care about what happens at two because a lot of times what you're going to find out is that we deal with limits where you can't get to that number cuz it for some reason it's undefined like our slope problem we just had okay so we're trying to figure out what's happening from the right and from the left and seeing if those are going heading towards the same exact value are they we're going from six down to four down to 4.4 where's this heading where's this heading it's heading towards four what we would say right now how you write it couple notes about this uh the function must approach the same value from both the left and the right so this is from the right this is from the left the function must approach the same value from the left and the right for the limit to exist if it does here's how you write the limit what you do is you put a little limb undercase which kills me because I can only write in capitals so you put the limit you put the variable you're working with you put that little arrow which I already told you is a approaching and you put the value to which X is approaching X is this one where are we where are we trying to get to on our X we're trying to find out what happens around the point xals 2 does that make sense to you so You' say the limit as X approaches two of your function what was our function say it again x s the limit as X approaches 2 of our function is equal to what did it what did the function approach here's what this says in English okay look at the board what does the function approach as the value X approaches two what's the function approach as X approaches two what's two that's the liit in general we have uh we have this this will be the last bit we had a limit of a function is equal to some number that's what capital L is that would stand for the limit of this function as X approaches a it's what the functions tending to do from both sides as X approaches one single value um the only thing that we we uh we need to know is the limit really doesn't depend on getting to a right X is never going to equal a X never gets a we just care about what's happening to the value of the function as X is getting really close to a from both sides up here on the board what's happening to the function as X is getting really really close to a from both sides it's getting really really close to four that's the idea of limit how many people understand the idea of a limit cool next time we work on how to find some limits all right so if you remember from last time we're talking about limits and what we're realizing is that a limit basically says or asks the question what is the function doing what's the value of the function doing as X approaches a certain number now do we ever care what happens when X gets to that number as far as a limit is concerned no no not really just what's the function doing where's it getting close to as we're getting close to that x value and that's that's the idea here so when we're talking about the limit of x -1 x^2 -1 that's our function we want to find out what happens as X approaches one now why can't I figure out what happens when x equals 1 tell me that your denominator zero that would be undefined right as a matter of fact it makes both a numerator and denominator zero and that would still be undefined uh we found out that that's is that a hole or an ASM toote do you remember a that's a hole that's going to be a hole ultimately this function has a whole so what we want to find out though is what's the function doing as we approach that certain value now one way we we figured out the only way we figured out these limits is to make up a table and this is kind of the elementary way that you discover limits right when you first learned now towards the end of the day I'll teach you some better ways on how to do this for right now though I want you to see what happens with the function with the limit so when you are finding these and your homework is going to ask you for that find the the limit of this function by making a table of when you do that you have your x value on the the top you have your FX values on the bottom in this case x -1- 1 and you start with the number you want to find the the limit of the the number where that you want to find the limit of the function where where X is approaching that certain value put that in the middle so for instance you're going to put one right there and really I don't want to know what that is because you can't plug it in anyway what we want to find out is what's happening from both sides is it going to the same value you follow me on this so what you need to do now put these numbers in order because it is a number line you don't want to have like I see a lot of mistakes on this I see a lot of people do this okay well I'm going to start at two and I'm going to go to 1.5 and then I'm going to go to 1.1 then I'm going to go to 1.1 right do you see how that's the wrong way to do it that that's going to show you One Direction but it's the wrong direction you need to have these numbers reversed so that this is just a little bit past one we're coming from the right hand side so if you want to find out where it's coming from don't don't have two right here have two over there somewhere if you're going to do that all right so these numbers that we have next to one they should actually be the numbers that are close to one all right not the other way around that are you following that now you have you're okay with that so we don't want to have the the smallest numbers over here that are the ones that are closest to one over there we want this just like a number line would look so here maybe we do start with 1.5 1.01 1.001 make it really close to one now the other way I'll probably want to start with 0 five right I want to make it symmetrical at least a little bit so where's 0 five going to go is it going to go by the one or over here on the left hand side left side left hand side I want the numbers are really closest to one well the closest to one that's going to show me the trend so 0.5 May be99 and 999 what I'd like you to do right now on your own take out your calculator find those numbers uh how about this the people to the see this is your left the people to the left side of the camera do these ones okay people to the right side of the camera do those ones can you do that for me so plug them into the function tell me what you're getting out I know there's a way to do this on t have you started to find those numbers does anybody have uh this one yet 66 continu does everyone get that one same one 6.66 so 6.67 how about the. N9 D we find that one 0. N9 52 can I get a double check on that 5 how much3 5 how about for this one that's that right one let's double check that one 099 minus one that should be pretty small you know negative something uh one more double check what' you say it was at five and then the first one 66 66 is your calculator scientific notation by chance no once you guys figure out your stuff all right I'll come back to y'all this side try this side can you give me 1.5 how much4 can you give me 1.01 497 how much 497 497 0497 take it okay can you give me uh 1.001 please 1.1 499 499 okay H interesting I'm guessing the six was probably not not all that accurate let's try that again was scient ah so 6 okay give me N9 what 503 give me n99 52 now do you see the the trend to which this function is is is getting close to can you see it what number is it getting close to from the right what number is it getting close to from the left that's why you put it in this order so you can see where it's approaching from the left and from the right if it's the same number if this is getting to 0 five and that's getting to 0 five you'd say okay that limit exists and that limit is 0.5 are you following me on this so the the limit right here would be5 and that's how you use a table to figure that out now of course this was a pretty mind numbing task right who wants to plug in 1.001 and do all this for for a whole bunch of numbers do you want to do that no I want to do that either because it's boring right we're going to find some better ways to do these limits in the next section for right now though I need you to get understand what a limit is a limit asks what's the function the value the function do as you get closer as your x value gets closer to that what's the function's value do what's the functions value do as your x value gets closer to that from both the left and the right if it's to the same number the limit exists and it's that number right you okay with this so far now are there ever cases when the limit doesn't exist for a number I want you to look at this all right we're going to talk about something called one-side limits and the question is what's the limit as X approaches to of FX now what we've had on the board already is what's called a right-sided limit and a left sided limit you have this kind of intuitive idea right in order for a limit to exist it's got to go there from both sides does that make sense to you it's got to get there from both sides for limit to exist so from the left and from the right has to be the same number I want you to look at this thing which way would be from the right over here or over here option one or option two one from the right would be this way true we can actually have right and left sided limits so here's how you write that a right sided limit in our case it would be a limit of our function as X approaches whatever value you're talking about in this case we're talking about the value two do you guys see where the two is coming from in this case we want to find out when X approaches two and how you say from the right hand side is you put a little superscript plus that means from right so in general you'd have this you'd have a limit of FX as X approaches a some number that I give you from the right that says a right sided one-sided right sided limit not your have be okay with that so plus means from the right you all all right with the plus meaning from the right what do you think a left side of Li is going to happen oh you guys are geniuses exactly yeah Left sided limit would be a limit of f ofx sure as X approaches a certain number in our case our certain number is two and that little minus or negative is from the left in general we'd have the same situation only from the left so let me ask you a question can you find the limits both right sided and left sided of this function let's try that together here's what the the question I'll go nice and soow so you can understand it here's what the question asks what is the value of the function as X approaches two from the right for right side and from the left from the left side okay so follow follow the function law I need you to tell me what what the height of my finger is what's the height of my finger when I get really close to this value along this line so this is from am I going from the right or the left in this case from the right from the right okay that it's always from the it's not to the it's from the right so from the right the height of my finger is okay I'm a little above a little above one what's the height of my finger going towards as I slow down I'm slowing down what's the height of my finger trying to get to it's trying to get to one do you see that it's not that my finger's getting close it's not the the limit's going to be two it's saying what happens to the value of the function the value is your y AIS what's happening to the value of the function or the height of my finger as my finger is approaching in this direction the number two does that make sense to you so here the height of my finger is trying to get to one as I'm approaching two from the left the right that's what I said from the right I'll edit that out you won even know beep just kidding I'm not going to edit that it's going to be on there forever yeah so from the right so from the right it's going to be approaching one yeah thanks for that you okay with that being one it's either yes or no if you're not that's okay but I need to know I need to have a question from you if you're not are you guys okay with the height of that being one as this function's approaching an x value of two so basically you're asking what's the Y value when you reach your x value okay okay we could see you what's your yv value when you reach your x value so now we're going to do from the left hand side what's the height of my finger aka the Y value as I approach the x value of two so what am I doing where am I getting close to as I'm getting to two from the left hand side what am I getting close to negative 1 yeah absolutely now here's a little note here's a little note for you in order for a limit to exist at a point the left side limit must equal the right side limit if it doesn't then the limit doesn't exist does that make sense to you it's got to go to the same place otherwise limit doesn't exists you can be one-sided sure here here but if it's not the same spot then you'd say overall the limit the limit means from both sides the general limit means from both sides so if it's not going to the same thing that limit doesn't exist so let me write that out for you and I'll explain it for a limit to exist at a point we'll call it a for a limit to exist to a in other words for this to happen for you to be able to say the limit as X approaches a is some number for that to happen you must have this the limit of x f ofx as X approaches a from the left must be equal to the limit of f ofx as X approaches a from the right it's got to happen so let's see if that Happ what this says in plain English is this you ready for the plain English part here's plain English the function from the left and the function from the right have to have the same value basically they've got to meet up somewhere you with me on that so let's check this out does the function from the right and the function from the left meet up what do you think do those lines meet up they do they meet up they come to the same point same value same height well let's see this limit was one this limit was negative 1 is one the same as negative 1 no so does the both the one-sided limits have the same value no you'd say this limit does not exist so we'd say the limit as X approaches 2 of f ofx does not exist this says why this says the limits from the left and the limit from the right must be the identical number you remember this example look up here on the board remember this example he you said the limit existed because that's 0 five and that's five basically it's coming to 05 you with me on that what if this was 05 and that was seven would that be the same number now this one's one this one's negative one is that the same number that limit doesn't exist okay it says you have to have the same function value on both the left and the right both would you like another example of this sure yeah why not as well have time let's practice okay let's let's get some good practice if you're not getting it I need some questions out of you because you really do have to understand this before we go any further so here's the idea in order to find the limit at a certain value in this case we're going to talk about two again do you see why that would be the interesting case here I want to break it down to you all right if I asked for the the limit of three it's not very interesting because look it here's the value of three does the limit exist at that point answerers clearly yes at three at three yeah look it the function's there the function's there the function's there it's all the same do you get me on that that's not an interesting example that's boring that just says the function is completely there therefore the limit must exist from the right and from the left you're coming to the same exact point do you get me on that that's not interesting if I ask you about two though that's more interesting you need to be able to find the limit from the right and the limit from the left as we're approaching an x value of two in math that means this can you do this can you do this and can you determine this let's see if you can let's see if you can I'm going to give you about 5 Seconds between each each problem to see if you can get it okay so right now I want you to determine on your own can you tell me don't say out loud okay let everyone do this can you determine on your own the limit of G ofx as X approaches 2 from the right do it on your own don't say loud give up five seconds the question in English asks this what is the Y value when X gets really close to two from the right that's what the question asks what is the function value or the Y value when X get really close to two from the right well that was like four people answering do you all not know it or do you know it know it if you don't know it then that's fine but you need to tell me if you know it then say it uh can you find out that the function value the Y value is about three or is getting close to three when X is getting close to two can you see that cuz you're following this along right you're going okay what's happening as I'm getting close to the value two along the X the Y is getting close to three so this is how much okay now I want you to do this on your own don't say loud again uh find me the limit of G of X as X approaches two from the left write it down in your paper don't say it loud from the left basically in English that says what's the Y value when the x value is getting close to two from the left starting from the left what is it one one one did you all get one how many were able to find one good okay here's how you tell if the limit exists or not here's how you tell you look at these ones are they the same you sure yeah one's 3 ones1 I mean how how much more sure can you be if those aren't the same does that exist if those are the same does that exist yeah I want you to know something really cool too about limits does it matter at all that that point was right in the middle of nothing we don't care about that point we don't care at two We Care what's happening to the function as we're getting close to that you see the difference I don't care about that it could as well not even be there it doesn't matter we don't care about that point we care about the function value the Y value as we approach the x value so if this does not equal that does that exist you put D you can really see it though can't you if the if the lines don't match up it's not there it's not going to happen the limit doesn't exist let's do one more and we'll call a good on this stuff um here we go for okay last picture last picture for us I want you to try to do that one completely on your own okay this is H of X I want you to find the limit of H of X is XT Clos 5 from the right I want you to find the limit of H of X as X approaches 5 from the left and then I want you to determine whether or not that limit exists at xal 5 you should you follow the questions okay go ahead and do that see what you get for oh oh let's see if we did it right okay so the limit of of our function as we're going from the right as we going from the left and then we're going to compare those numbers so from the right means from the positive or from the more positive I should say because sometimes both positive from the left means from the less positive or negative whatever your case may be so from the right from the left from the right what is the Y value as your x value approaches five what are you getting towards okay well what about that one does the function actually approach that approach it see that's a difference we don't care what it is at that number we care what it's approaching as we're getting close to that number you see the difference right okay so you all got how much perfect let's do it from the left from the left what's the the Y value getting towards as your x value is getting towards five good is the right side of limit the same as the left side of limit does the limit exist absolutely yeah why don't care what happens to that point just because this point doesn't fill in that hole it doesn't matter as long as the functions values go to the same exact spot from both sides do we care what happens when we get there no we don't care about that we care about this that limit is three how many people have a better idea about the idea of a limit right now so given a a graph you can do this okay cool hey let's talk about that for a second what's the one number you can't plug into that Z what would happen if we discovered what what the function does as we approach zero isn't that an interesting question it's kind of cool kind of what happens to the function as we get close and close to that let's talk about that for a second now in order to talk about that we need limits what's the function do as we approach that so just like we did over here we're going to talk about a right sided and a left-sided limit but we're going to do with a table now what number are we approaching I'm sorry what was it where's the zero go on our table to the left to the right or in the middle because that's the number we're trying to get to are we ever going to be able to get to the zero now we can't plug it in because then it's undefined that would be a problem so now let's pick some numbers that are getting close to Zero from the right let's start at like oh I don't know 0.5 and we'll work our way down so 0. five is 05 supposed to go here or here the right or the left so five all right 01 and 01 would you agree those are numbers that are pretty close to zero right uh what are the numbers to the left of zero what do you want to use the same ones but negative okay good they have to be negative though because they left to zero so5 that'll work And1 And1 so let's use those numbers again left side people do the left side right side people do the right side these one should be a little bit easier to plug in yeah little bit by the way this is going a little bit out of the scope of this section at this point we're talking about something a little ahead of time uh it's interesting to me so we're we're covering this right now it will come back at us uh later on right but I'd like to make sure you see it in limits at least once or twice before we get to that section so if you take one and you divide P5 I'm hoping that you got two did you get two and over here you probably got well I'm going to be a genius about that -2 did you get2 genus no so okay good I scared myself for a second so two and-2 very good uh now how about 01 if you divide one by 01 you should get let's see move the decimal place you should get and I'm going to hope this is 00 yes if you divide by 01 that's let's see 1 2 three decimal places you're just dividing one by decimal places there ending into one that should be let's do a little critical thinking here all right first question is is this number getting bigger or smaller as we're approaching Zero from the right is it ever going to stop getting bigger 1 divided 000000001 is going to be really really really really really big right and it's going to be positive how about this way is it getting bigger or smaller way smaller and it's never going to stop getting smaller it's not going to a certain number now first the second question is does the limit exist are they going to the same exact number No in fact as we're going from the right where would you say this is going because if you draw the picture it's going like like that it's going where INF Infinity if you divide by smaller and smaller numbers that are approaching zero keep dividing by 00 forever and then little one you're going to get really really huge numbers right and you can keep getting smaller and smaller still so what we would say is the limit as X approaches zero from the right of our function which is 1 /x is positive infinity and you you technically don't have to put the plus because Infinity without that plus still means positive Infinity but I want to show the difference limit as you approach it I'm sorry X approaches zero from the left is what would you say that would be is negative Infinity the same thing as positive Infinity so limit clearly doesn't exist in fact you can see this from the graph if you graph one of Rex your calculator whatever you you graph it on look like this that's it you can see as we get towards zero it's skyrocketing as we get towards zero it's really going into Abyss right so it's it's not going to be the same thing but this leads us to a couple ideas whenever we have a limit I'll write this up for you if you have a limit as X approaches a from either the right or the left of some function and you figure out that it's positive or negative Infinity say it again if you are approaching a number right from either the right or from the left and it's going to Infinity either positive or negative what that does for you is it gives you an what's this thing called gives you an ASM toote gives you an ASM toote because you say okay if we're going towards Infinity as we're reaching as if my y Valu is going to Infinity as my x value reaches a number that means I have to be shooting up if my y Valu is go to negative Infinity as I reach a certain number that means I have to be shooting down so one of those cases it's going to be some sort of ASM toote you're never going to actually get to that point you can't you can't so uh there are really four cases I'll draw them over there so we see them four cases of of asmp tootes and the relationships to their limits for so the one case is what happens if we approach a from the right and we're going to positive Infinity another case would be what happens if we approach a from the right and we go to negative Infinity another case would be well what would happen if we went we did the both rights right the right if we went to a from the left and went to positive infinity or what would happen if we went to a from the left and it gave us negative Infinity would you agree those are all four permutations all four cases of of this in either case we just talked about this if your function is going to positive NE Infinity when X is going to a from the left or the right you're going to have an ASM toote so every one of these is going to be an ASM toote well that's the worst ASM toote I've ever drawing glad to experience second not let's kind of diagnose what these things would do I want you to really think of what these limits should do to our our function here okay now if I say x is approaching a from the right should I start here or start this way am I going this way option one or this way option two option one so if the function's approaching a from the right and it goes to positive Infinity should I be going up or should I be going down so this would be this type of graph it can do whatever it wants over here no problem but when it gets to here it shoots up this is not going to be this isn't going to be told by our limit as we approach a all right has nothing to do with it it's what happens as we get close to a it's going to be going towards positive let's let's talk about this one now I know we were just coming from the right hand side this one's going to be going well not up not positive Infinity it's going to be going down negative Infinity do whatever you want over here but it's going to drop it's going to be ASM totic to that that line how about this one can you picture what that one does in your head already draw a graph if you want to right now you should be able to draw something with that are we going option one from the right or option two from the left from the left and should we be going up or down which one it's going to positive Infinity up or down up lastly the only case we have left is from the left going down to negative infinity looks something like that would you raise your hand if you feel okay with what we've talked about so far today all right that takes care of kind of our introduction to limits we talked a little bit about that now for the rest of our day I'm going to teach you how to compute limits because if you had noticed this this isn't all that fun I'm going to show you some better ways on how to do that would you like to learn that cool any questions before you go on the same ex the limit would exist yes it would so if we have a one side limit this way and want that limit that way and they both go to positive Infinity yeah sure we'd say the limit is positive Infinity because it goes from both sides or if the limit is going both negative Infinity that would exist so let's see this way yes this way yes this way no this way no it takes practice to do that by the way I showed you that already just one question about this one yeah um it's approaching from the right should should approaching from the left this one no the one above this one this is approaching from the right going to negative Infinity oh I see the plus now oh never mind yeah you know what let me write that out too um if this and this do you see the this the thing here if we go from the right and from the left and they both go to positive Infinity then this was your question yes sure if the two if this works all the time if the left side limit and the right side limit go to the same thing the limit exists uh you could substitute in if this is going to negative infinity and negative Infinity that would be negative Infinity so you can draw that that Corr as well good question any other thank for that one any other questions before we continue all right let's compute some limits you got that