Calculus 1 Lecture 1.1: An Introduction to Limits

so let's get started with our our actual calculus stuff always gonna be awesome so with our calculus we're gonna talk about limits first now our limits is really the basis to our calculus it's how we do calculus we're gonna find out some kind of tricks to calculus later but for right now if you know how to do limits you're gonna 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 for right now I want to lead you through an introduction to calculus and what it's all about are you ready to learn that so review is over with thankfully now we're going to slow down just a little bit we're going to talk about actual calculus stuff so no more really hair-raising kind of oh my gosh so much stuff we're gonna go a little bit slower but it's going to be more in depth you ready for slightly slower not not much look in calculus we have two basic goals in calculus one this is in our course calculus so here is your goals goal number one the first goal one that we're going to spend most of our class on least the first half of our class on is this school giving any curve giving the curb not straight lines real engine easy but given a curb I want to be able to find the slope of a curve at a point isn't that the interesting idea I said 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 okay I'll find you slow let's see slope is one-half is that accurate no idea but well 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 going to 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 one is to find the slope of a tanner to find the tangent to a curve at a point the tangent is the line intersects a current one and on one spot 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 run calculus fast cops the best place 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 to the tangent and and vice versa tangent and slope go hand-in-hand for the thrust goal number two which is the last half of our class maybe a little less than that goal number two is that's also a really interesting question so cool by the way this stuff is stuff you can't answer in any other class right you can't you can't do that with algebra there's no way you can do that algebra you can't find a slope of a curve be kidding me so the straight line easy simple curve wow that's it is that not an intriguing idea to you it should be if you're in this class he's like that's good it's kind of cool I can find some of a curve calculus the next question is probably even a little bit more interesting next question is let's suppose I have some funky any curve digit as you can think of as long as defined by a function can you between two points find the area under that curve can you do it an answer is right now with what you know can you do that with geometry because I could say okay great find the area of something that's curving can you do that no no you can't do is use the area for rectangles on a rectangle can use area for triangle that's not triangle you can't break this up in any way that you have geometric figures can you not even circles not even with radius is because the radius is changing so the next question is can you find the area under occur 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 rights and crazy questions going on here and amazingly it's actually not that hard this is I think it's easier precalculus it's easier than that you know I'm different not easier but if you don't freak out this is not sus reaching what we're going talk about first we're talking the tangent problem I'll give you a very intro to this but I'll talk about later on we're going to talk about this for most of our day to day so let's talk about the tangent problem are you ready do you have any questions on the ideas of how 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 find the area under a curve between two points very cool ideas so let's work on the tangent problem left neck you're lucky you're here today okay this is this this is like the fundamental idea of calculus is so cool or to figure out what a limit is from doing this idea right here to here's the tangent problem the tangent problem says if you give me any arbitrary curve so I'm just making out a note 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 do you get the idea about the tangent problem right now right slope in that curve at that point because if we have the slope and we have the point we can make the equation of the tangent you give me that's that's pretty easy that's just Y minus y1 equals MX - x1 so right now we have the point what we're worried about is the slope of the curve at that point you follow now the premise is if we're going to make a tangent line how many points do you need to make up a line how many points do we have that's the problem all 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 problem so here's what we're going to do where it's okay well let's let's put some points somewhere else on this thing how about cute 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 secant line if you've had some geometry is secant connects two parts of it will usually a circle but it 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 agree that we need two points proof or an align 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 a 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 cube I can't move P but I can move the point Q can you move Q to make this secant 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 is interesting so we're going to find out what happens as we move Q down this line so if I knew Q here is that better or worse better because I'm gonna have these different colors shoot that this would be like Q sub 1 maybe a different point for Q if I move it closer to give you a better or better okay if I move it closer it's going to be even better if I move it closer it's going to get even better 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 because then you'll live one point how many points of your line - so can you move it close the answer is sure we're doing it right now can you have it close yes can you be 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 line that's great that'd be fantastic that Tsar ID 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 speed by the way really really really really that means very very very very and we kind of look 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 let's move a little close not really far away that's a Stuckey approximation well if you look really close through that stinkin 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 paying you 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 what we just talked about this we just said this um that there's nothing new as Q gets closer to P the secant line more closely approximates the tangent model now have you understood that the first okay so as Hugh approaches I may use that word we're going to use that word limits as well approaches that means gets closer to as Q a Pope approaches P we get closer to a tangent or other words in a secant more closely approximates the tangent group 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 ah let's move to B because there would be a difference okay there wouldn't be two actual points at the same point twice and that's that's problem could make an equation with line with that so one little note we can't just let you go P because we need two points to make them on you need two points that would be a problem here's the idea of a limit alright this is a big big idea here's the big big idea from 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 the midpoint can you find some space between it so we've over the midpoint can you find some space between it can you keep doing that for eternity between 82 points there's there's a gap in that gap you can fill it in with another point because points have no breath that means that even though we draw them on the board like this they really don't have a distance between them so two points can get what you agree infinitesimally close to each other then he's so so close that you can't put anything between them but there's still say little distance and then you can get even closer that and closer that and closer would you agree that to that so here's the idea behind calculus it says we can get hue 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 get it so close but 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 there are different points but they're so close doesn't matter that's the whole idea so the the big picture for our limits is if we let you get really really really close to P the secret will be to a tangent let's save it or approximate it so closer than it is a medical really really really if we like you get really really really close to PL use up your toes now these are more mouthing word for that just a minute really really close to P the secret will be identical to the tangent here's the idea right here here's the idea now we 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 close without actually getting here that's the idea of a limit you kind of understand the idea of a limit we would say that Q would be a limiting position that means we've got it all the way really as close as we can to P but it never touches P let's say I did a limit it gets really really close so the idea of really really close is a limit in the most simple terms so this is it would you like to see an example of how to actually do this the be interesting to you how to find the so computer one you'll see that it's kind of fun okay we're going to work through this I'm simply doing this to give you an understanding of how limits work and the jump that you can make okay 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 in calculus I'm going to make one limit jump in this problem show you what it is but this is stuff that you could actually do we're just going to work and use calculus to do it better you ready for do it more mathematically before I go any further did you guys understand the tangent problem not sure how if you're ok with the tangent problem you understand I did my limit moving to point really close to but never actually getting there again why can't we get there two points yeah I didn't know okay cool now we're ready outside first a little step off into confidence 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 let's go over to one one and let's put a point there so P is the point 1 1 I'm going to walk you through how we're going to do this right now ok 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 piece set in stone you're not going to move feet what other point do we have we're going to make point Q key you what's the coordinates for Point Q points tell me its coordinates for point Q and we can't use actual numbers right that would that would be very good so I know 2 4 would be on there and three times a on there for 60 and so on but I'm going to use actual points in general what are the points for what are the coordinates for any point XY very good ok x1 so now write down X Y and have your race are handy have your racer handy because I'm not going to have X Y I want to keep this in terms of one variable now using your knowledge of what the function is what's the function y equals blood which city and y equals what where y equals so instead of having X comma Y would it be okay with you if I'd add X comma x squared yeah yes no because Y is x squared right so matter so erase that and I want to predict squared cool all right now we have this secant line going from P to Q P is a fixed point 1 1 Q is a moveable point x comma x squared would you agree that any point on this line will have the coordinates x comma x squared no matter what I plug in right so that that moogle point that's going to have the same chord as 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 minus y1 equals M X minus 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 minus 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 a tangent line and then X minus x1 now the cool part about this 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 how do we have 1 1 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 gonna try to find right now the slope of the tangent are you ready for it be sure we're going to use the idea of slope of a secant so we're going to take a break right now we're going to come back to this in case you're just a little bit lost here's what we've done so far this is y squared I gave it to you we fixed one point according to my problem that's one one we've made up another point a movable point Q q is that X comma x squared with x squared come from that's because it's Y all right great we now know that the equation of a line is this so therefore the equation of a tangent line would be the same as that equation it's just we 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 seek it and make it into a tangent line by moving cue really close to peds you get the idea basically doing that with this problem so let's take a look at a slope of a secant our specific secant hey by the way do you remember the slope in general the slope formula slope is 1 over 1 right okay great in terms of our coordinate slope is Delta Y over Delta X specifically in terms of our coordinates what is our slope line that's what I'm looking for y2 minus y1 over x2 minus x1 otherwise you get the negative so that's look so y2 minus y1 x2 minus x1 do you follow I'm gonna feel okay with it so far just look formula what we're going to do we're now going to find the social part of secant we've got the secant on the board right the secant will be the slope and so the slope of the secant will be the slope from points P to Q agree let's plug in these coordinates we only have a board plug in these quarters into that formula in our case our y2 is what our y2 and the x squared what's our y1 we won what's our x2 what's our x1 okay y2 minus y1 over x2 minus x1 notch if you okay with that so what we have for the so far secant is y2 minus y1 over x2 minus x1 are you okay with this so far find out where the silk comes from there's a couple notes I want to make at this phone first I'm reiterating a lot of stuff here I know that I am I'm doing on purpose because 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 cap 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 know what they are right now if you don't understand what you're doing it's irrelevant you're just now just doing formula as it sucks if you know what it comes from what you're actually doing 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 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 to agree as Pete I'm sorry XQ 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 let's over here right now you agree with it already how high reaching 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 who's seeking these close to the tangent how close they 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 agree it's kind of neat right school idea now that the problem is here's your snoke number one here's no number two okay this is interesting right where are we trying to find the slope what point one one well and what's the x coordinate of 1 1 to find the slope of any where you plug in that value right this gives you the slope of the secant line true plug in 1 look at one what happens know you know gets here oh you get zero over zero over zero is not zero that's a big problem zero over zero zero over zero is undefined right isn't that an issue this is why folks this is the reason why why you cannot have the point getting you can't get Q close to P because if you plug in one it's undefined that means that you're you have the same point so you can't let x equals one 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 it will have one point right you can't find the slope it fails it's undefined at that point this is yqk will be if it does if try find slope of the secret cat at that point move Q all the way down and plug in that x-coordinate if it's the same thing you didn't get something undefined you should try to find the difference between points that don't exist a difference that doesn't exist and that's going to be your divided by 0 there's no difference in x axis that would mean your bottom is 0 you something undefined you undefined slope that's a bad thing so we can't let Q be 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 blown like a mind grenade you're seeing yes man once yes man so fun had a ripple deliverable it's gone triple ruff you're not seen it that just sounded really stupid watch yes man it's pretty funny now so it's next part good blow Iman can you factor that in fact money what we knew about acid and stuff like that you know that you have zero zero is factorable because of all some stuff in mathematics that I haven't told you it yet but if you ever have the number that makes zero it means that that's factor if you have some number the same no that makes two polynomials zero it means they have a common factor we've got a common factor here in fact if you were fabulous this is going to give you X plus one X minus one over X minus one degree okay do you think see anything that simplifies out of that haha 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 X equal one is Q actually getting to the point P no so is X actually equal one now we're getting really really close I'm talking like one point zero zero zero zero zero 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 don't we're not really altering the domain whatsoever because we already knew X was an equal one so we have this little restraint already X isn't an equal one so then what we know is okay the slope of the secant now equals x plus one were there any questions on this because I got to erase it you have anything 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 not eliminating the main 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 what happens as cuties closer 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 416 so extra now before plug in for dinner how much would you get okay now let's move it down to 3 plug in 3 here where was you move it down to 2 how much you two you can move it down to one point five two point five moving down to one point three what would you get moving down to one point one what would you get moving down to one point zero one what would you get two point zero one good move down to one point zero zero zero zero one what would you get move down at one point zero zero zero zero forever and then a little one at the end what would you get two point zero for every little one would 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 make sense to you so our secant to be really close to visual that says this lets us make the jump this is a limiting position says that we know the limit of the slope of the secant lines - what that means for us is that the smoke of the tangent line actually is - 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 secant line gets really really close to two now I can't let X equal one but I can make the jump that if I could let you look at X equal 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 the C can attain so because the sequence 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 I'm sure we know that it would be Y minus y1 which is one equals the slope of the tangent hey we know it's 2x minus one now of course we don't generally leave things in point-slope we'll solve them do you see where the 1 the 1 and the 2 are coming from folks our point is 1 1 our slope is now - that's a slope of tangent if we solve that we get y -1 equals 2x - 2 add the 1 y equals 2x - 1 that's the slope of the curve at a freaking point that's awesome oh my gosh it met cool we spend the first tangent line to a curve that you've done now granted it's not very hard curve but here's what this I'll show to you you can actually graph it here's minus 1 it crosses there you go up to 1 2 over 1 oh that's the point as I graph it the intersects it only one spot that's it that's the tangent line isn't it awesome your mind should be blowing is that why you're looking at me stunned like yeah that you're stunned face it should be your stunned face ok now by show of hands after 40 minutes of doing one example how people should we talked about yeah do you have down right 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 how we're going to do things differently when we actually get to the calculus we study limits more first but we will get there now the next thing we got to talk about is the different problem I'm going to I'm not just spend a whole lot of time on I'm basically just going to introduce it to you because I know for sure you guys are not going to remember this by the time we actually get to area under curve so I'm going to 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 air problem says can you find the area under curve between two points answers 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 firstly then 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 what maybe would be an idea oh I couldn't yes sure if I did this and say okay from here I just want to make a rectangle would that be an approximation would 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 and a rectangle from here here then one from here here and one here 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 wide rectangles well let me ask you can you find an area of a rectangle that's why rectangles and they're easy to draw as well so if we make lots and lots and lots of little bitty teeny bitty mini 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 where as 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 add them up it's going to involve limits because we're letting some number go to infinity you can't reach infinity but a limit will take care of that for you that's the idea behind the very 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 what I you do a problem we'll do that later chapter 4 some like that so let's define a limit what a limit says in English is what does the function do as a variable approaches are given done that's what it says in English so minutes 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 your zoning there's no no can't have one one online we'd also make a undefined point right again to find that would be bad but we saw what happens we get really close the function we've got really close to something else that's any of a limit so we don't care what happens as you get to the value what happens is you approach the value of for us this is this is exactly what we did actually kind of in our heads what happens what happens to x squared as X approaches let's do two that's X approaches 2 I get really English but that's just invalid what happens as X approaches 2 now do we care what happens if you plug in 2 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 of X I say I'm trying to get to 2 I don't care what that value is what I'm trying to see is what the function is doing is I'm getting close to that thing so what's happening as we approach it from the right what's happening as you push from the left here is oh gee I already ruined it give me some numbers to the right of 2 two numbers to the right 3 maybe something smaller than 3 like 2.5 that's what I was about to write ruin it give me something a little bit closer to closer than that baby 2.1 yeah which you point 0 1 we want really close right so two point one and maybe two point zero zero one that's pretty close take your carriage up that 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 keep it kind of symmetrical 1.5 I like that one will do uh 1.9 all right and 1.999 I love that that work for you take those numbers our functions x squared let's look what happens to the function as you plug those things in if you plug in 2.5 what do you get when you guys with a six point two five or something like that plug in 2.5 ba 6 4 2 5 okay so we'd start with six point two five what we care about is just this side look at the delimiter 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 the limits now this is a very easy example we just were working with x squared we know the answer is going to be 4 right whether it should be 4 we're going to see if that actually happens when we when we evaluate our limits plug in 2.1 someone out there telling what you get when you plug in 2.1 would you for my balloon okay that's close to 4 tell me what you get when you plug into point zero zero one two point zero zero what was it for Jim zero four zero zero one okay like that how about one point five let's that's 225 in it how about 1.9 plug in one point nine what do you get to do that x squared three point and now do one point nine nine nine one point nine nine nine what you get how much any point nine six 3 4 9 6 0 mom oh yeah what's going on let's see with a limit we really don't care about what happens at 2 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 because it for some reason is undefined like our slope probably just had okay so we'll try to figure out what's happening from the right and from the left and see if those are going heading towards the same exact value are they we're going from 6 down to 4 down the 4.04 where's this heading toward floor where's this heading it's heading towards for what we would say right now beside you right it couple notes about this the function must probe 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 to live it what you do is you put a little limb under case 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 approaching and you put the value to which X is approaching X is this one where are we trying to get to on our X we're trying to find out what happens around the point x equals two does that make sense to you so you say the limit as X approaches two of your function what was our function sit again x squared the limit as X approaches two of our function is equal to what did it what did the function approaches or the 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 listening to that's the limit in general we have we have this this will be the last day we had a limit of a function is equal to some number that's with capital L is that would stand to the limit of this function as X approaches a it's what the function is tending to do from both sides as X approaches one single value the only thing that we we need to know is the limit really doesn't depend on getting to a right X is never going to play that's never to say we just care about what's happening to the value of the function as X is getting really close to a for both sides up during 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 it everyone how many boys have I given limit cool next time we work on how to find some limits alright 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 the limit is concerned no not really just what's the function doing where's it getting close to as we're getting close to that x value that's that's the idea here so when we're talking about the limit of X minus 1 over x squared minus 1 that's our function we want to find out what happens as X approaches 1 now why can't I figure out what happens when x equals 1 tell me that that would be undefined right as a matter of fact it makes both the numerator and denominator 0 and that would still be undefined we found out that that's is that a hole or an asymptote you remember that's a hole that's gonna be a hole but ultimately this function has a hole so what we want to find out though is what's the function doing as we approach that certain value now one way 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 learn now towards the end of the day I'll teach you some better ways on how to do this but 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 it a plug when you do that you have your X values on the top you have your f of X values of the bottom in this case X minus 1 over x squared minus 1 and you start with the number you want to find the of the number where that you want to find the limit of the function where X is approaching that certain value within 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 well 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 a two and I'm going to go to 1.5 then we go to one point one then when we go to one point zero zero zero one but do you see how that's the wrong way to do it 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 to right here have two over there somewhere if you're going to do that 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 but are you following that not sure if you're okay with that so we don't want to have the smallest numbers over here that are the ones are close to one over there we want this just like a number line look so here maybe we do start with 1.5 1.0 one one point zero zero one make it really close to one now the other way I'll probably want to start with 0.5 right I want to make it symmetrical at least a little bit so where's point 5 unit goes you can go by the one or over here on the left hand side left hand side on what the numbers are really closest to one well the closest to one that's going to show me the trend so 0.5 maybe 0.99 and point 9 9 9 what I'd like you to do right now on your own take out your calculator find those numbers halep is the people to the C this is your left the people to the left side of the camera do these ones ok people to the right side of the camera do those ones kgf when we so plug in the function tell me what you're getting up the way to do this have you started to find those numbers does anybody have this one yet 6.6 does it one got that one say 16.6 666 per so 6.67 now at the point nine nine they would find that one point nine nine okay I get a double check on that five point how much point five point close to the elbow for this one let's look with our one right one let's double check that one point nine nine minus one that should be pretty small even they do something one more double check what you see it was point six it is your tabular scientific notation by jeans once you guys figure out your stuff all right I'll come back to you on this side try the side can you give me one point five one how much fun four point four can you give me one point zero one point four nine seven how much 0.4 97 0.107 okay can you give me one point zero zero one please one point zero zero one point four nine nine okay mmm interesting I'm guessing the six was probably not not accurate let's try that again ah so point six okay give me point nine nine point one point five zero three give me point nine nine nine point five zero zero two 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 you from the right what number is getting close to from the left that's why you put 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 0.5 and that's 0.5 you'd say okay that limit exists and that limit is 0.5 you follow me on this so the limit right here would be 0.5 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 one point zero zero one and do all this for for a whole bunch of numbers do you want to do I want to do that is this boring alright we're going to find some better ways to do these limits in the next section for right now though I need you to understand what a limit is a limit asks what's the function the value of the function do as you get closer as your x value gets closer than that what's the functions 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 if you locate with a so far now are there ever cases when the limit doesn't exist for a number I want you to look at this alright we're going to talk about something called one side limits and the question is what's the limit as X approaches 2 of f of X 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 - it's got to get different both sides for limit exists 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 over here option 1 or option 2 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 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 - do you guys see where the two is coming from in this case I want to find out when X approaches two and how you say from the right-hand side as you put a little superscript plus that means from the right so in general you have this you have a limit of f of X as X approaches a some number that I give you from the right that says a right sided one-sided right-sided limit none have you okay with that so plus means from the right you alright with the plus me from the right what do you think a left side of them is going to have oh yes geniuses exactly yeah left-sided limit would be a limit of f of X sure as X approaches a certain number in our case our certain number is 2 and a little minus or negative is from the left in general we have the same situation only from the left so let me ask you a question can you find the limits both right side and left side of this function let's try that together here's what the question I'll do necessary so you can understand it here's what the question asks what is the value of the function as X approaches 2 from the right for right side and from the left from the left side okay so the follow the function well I need you tell me what the height that 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 on the right from the right okay that's always from that's not to that it's from the right so from the right the height of my finger is okay a little bit more 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 let's try to get to one do you see that it's not that my fingers getting close it's not the the limits going to be - mm-hmm it's saying what happens to the value of the function value is your y-axis 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 that my finger is trying to get to one as I'm approaching 2 from the left that's what I said from the right I'll edit that out you don't even know peep just kidding I can edit that's gonna be on there forever oh yeah so from the right so from the right it's going to be approaching one yeah thanks for that okay with that being one it's either yes or no if you're not that's okay but I need to know I didn't have a question from you if you're not you guys okay with the hype of that being one as this functions approaching an x value of 2 so basically you're asking what's the Y value when you reach your x value what's your Y value when you reach x value so now we're going to do it from the left hand side what's the height of my finger aka the Y value as I approach the x value of 2 so what am i doing where am I getting close to as I'm getting to 2 from the left hand side what am I getting close to leave 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 a good of the same place otherwise the limit doesn't exist you can be one-sided sure here here but if it's not same spot then you'd say overall the limit but then it means from both sides the general limit means from both sides so 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 fate for a limit to exist at eight 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 of X as X approaches a from the left must be equal to the limit of f of X as X approaches a from the right it's got to happen so let's see if that happens whether it says it in plain English is great for the plain English part here's language 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've done that so let's check this out does the function from the right and a function from the left meet up what do you think do those lines meet up they do is they meet up they come the same point the same value same height let's see this limit was one this level is negative one is one the same as negative one so does the both the one-sided limits have the same value now you'd say this limit does not exist so we'd say the limit as X approaches 2 of f of X does not exist this says why this says the limits from the left and a limit from the right must be the identical number in this example look up here the board Booth's example hey except the limit existed because that's point 5 and that's point 5 basically it's coming to point 5 you with me on that whatever this was point 5 and that was 7 would that be the same number now this one's 1 this one's negative 1 is that the same number that limit doesn't exist ok it says you have to have the same function value on both the left and the right Oh we'll check another example this why not you let's practice okay let's let's see some good practice if you're not getting it I need some questions out here because you really do have to understand that's reporting awaiting 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 right if I asked for 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 answer is clearly yes at three at three yeah look at the functions they're the functions and functions there it's all the same do you get on that that's not an interesting example that's boring they just said 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 asked 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 2 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 five seconds between these problems and 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 turn on your own the limit of G of X as X approaches two from the right doing your own osela give it five seconds the question in English asks this what is the Y value when X gets really close to 2 from the right that's what the question asks what is the function value or the Y value when X gets really close to 2 from the right that was like 4 people answering do you all not know it or do you know it if you don't know it then that's fine but you didn't tell me if you know it then say it can you find out that the function value the Y value is about 3 we're just getting close to 3 when X is getting close to 2 can you see that because your phone is along right you okay what's happening as I'm getting close to the value 2 along the X the Y is getting close to 3 so this is how much ok now I want you to this on your home don't sell out again find me the limit of G of X as X approaches 2 from the left write it down here paper don't sell out from the left basically in English that says what's the Y value when the x value is getting close to 2 from the left starting from the left what is it 1 did y'all get one how many we 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 be sure yeah once three ones one I mean how much more sure can you be if those aren't the same does that exist if those are the same does that exist I wish you know something really cool too about limits does it matter at all at 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 that's what I'll even be there does it matter we don't care about that point we care about the function value the Y value as you approach the x value so this does not equal that does that exist there you could be you can really see it though can't you if the lines don't match up it's not there it's not going to happen get it do limit doesn't exist let's do one more and we'll call you it on this stuff there you go okay last picture less fish 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 as X approaches 5 from the right and once you 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 and x equals 5 or you can shoot all the questions and go ahead and do that see what you get you oh let's see if we did right okay so the limit of our function as we're going from right as you're 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 your composite 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 5 what are you getting towards okay well one does the function actually approach that approaching yes difference I 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 y'all got how much perfect let's do it from the left from the left what's to the Y value getting towards as your X values getting towards five very good is the right side limit the same as the left-sided limit yeah does limit exist yes absolutely yeah why don't care what happens 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 nope we don't care about that okay with this not limit history how many people have a better idea about the idea of a limit right now so given 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 what would happen if we discovered what what the function does as we approach zero isn't that interesting question it's kind of cool and it's true 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 once 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-side limit but we're going to do with the 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 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 well I don't know point five and we'll work our way down so 0.5 X 0.5 steps ago here or here the right or the left right right so 0.5 all right point zero one nine point zero zero one would you agree those are numbers that are pretty close to zero what are the numbers to the left of zero what do you want to use okay good they have to be negative though cursor to the left to zero so negative point five that'll work and negative point zero one and negative point zero zero one so let's use those numbers again left side people do left side right side people do right side it should be a little bit easier to plug in yeah 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 cuz it's interesting to me so we're covering this right now it will come back at us 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 divided by 0.5 I'm hoping that you got to did you get to and over here you probably got well I'm gonna be a genius about that negative 2 you get nave - no ok good who scared myself for a second so - and NATO - very good now how about point zero one if you divide one by point zero one you should get the C move the decimal place you should get unfunded and I'm gonna hope this is negative 100 yes if you divide by point zero zero one that's let's say one to three decimal places you're just fighting one by decimal places there and even 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 you are approaching zero from the writer is it ever going to stop getting bigger 1/3 0 0 0 1 is gonna be really really really really really big right and cb+ 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 to where finiti if you divide by smaller and smaller numbers that are approaching zero keep divided by point zero zero zero zero zero forever in the 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 0 from the right of our function which is 1 over X is positive and thin and you double you technically don't have to put the plus cuz infinity without that plus still meets positive and I want to show the difference limit as you approach I'm sorry as X approaches zero from the left here's what would you say that would be is negative fitting the same thing as positive infinity so limit clearly doesn't exist and cut you can see this from the graph if you graph 1 over X your calculator if you draft mountain like this that's it you can see as we get towards 0 its skyrocketing as we get towards 0 it's really going into a this right so it's it's not going to 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 so 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 get your NASA total gives you a something because you say okay if we're going towards infinity as we reach as as if my y-values going to infinity as my x value reaches a number that means I have to be shooting up if my Y values go to negative a as I reach a certain number that means that have to be shooting down so one of those cases it's going to be some sort of asymptotes you're never going to actually get to that point you can okay so there are really four cases I'll draw them over there let me see them for chasm of asymptotes and the relationships to their limits 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 for right if we went to a from the left and with the positive infinity or what would happen if we went to a from the left and it gave us negative infinity which agree those are all four permutations all four cases of this in either case we just talked about this if your function is going to positive infinity when X is going to a from left or right you're going to have an asymptote so every one of these is going to be an asymptote well it's the worst asymptote I've ever drawn second perfect it's kind of diagnose what these things would you I want you really think of what these limits should do to our art function your okay now if I say X is approaching aid from the right chefs are here or start this way am I going on this way option 1 or this way option 2 function 1 so if the function is 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 should 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 rightist that hasn't a clue it's what happens as we get close to a it's going to be going towards positive let's let's start with 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 phase we going down they infinity whatever you want over here but it's going to drop it's going to be at some product to that how about this one can you picture what that one does in your then already draw a graph if you want you 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 let's go to plus infinity up or down 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 talk about so far today all right that takes care of kind of our introduction to limits but we talked a little bit about that now but the rest of our day I'm going to teach you how to compute limits because if notice 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 any questions before you want the limit would exist yes it would so if we have a one side limit this way and one side them in that way they both go to pause and fin yeah sure we'd say the limit is positive in because it goes from both sides or if a limit is going both Megan infinity that would exist so let's see this way yes this way yes this way no this way now that takes practice to do that by the way yeah it's approaching from the right it should be a perfect on the left this one never one above this one this is approaching from the right going to negativity oh I feel a plus on everybody yeah you know what let me write that out too if this and this do you see that this is a finger if we go from the right and from the left and they both go to positive infinity and this was your question yes sure if the two if this works all time if the left-sided limit and the right sideline go to the same thing the limit exists you could substitute in if this is going to negative fifty at negative three that would be negative finish so you can draw that that corollary as well good question any other good point any other questions before we continue all right let's compute some limits you get that

Channel: Professor Leonard

Views: 2,475,244

Rating: 4.9153433 out of 5

Keywords: Leonard, Lecture (Type Of Public Presentation), Calculus (Concepts/Theories), Math, Limit, Calculus (Field Of Study)

Id: 54_XRjHhZzI

Length: 87min 26sec (5246 seconds)

Published: Wed Aug 22 2012

Reddit Comments

[u/J_hvnnen]

Professor Leonard ‘s videos are the only reason I was able to pass Calculus 3. One of the most excellent resources out there.

[u/wheezy1749]

https://www.youtube.com/user/patrickJMT

PatrickJMT is my recommendation. Clear, quick, straight to the point examples and teaching.

[u/thecountofjeans]

I'm still struggling with the most basic mental arithmetic

[u/lghtspd]

Prof leonard is great! MathBFF helped me a lot with algebra and calculus too

[u/Dear-Crow]

This is the dude that got me through calc. Awesome teacher!

[u/sansaman]

Meh. I prefer NancyPI for math.

[u/MyPusyTasteLikePepsi]

Anyone know a good channel for organic/biological chemistry

[u/bikerholic]

That Aussie bloke Eddie Woo Calculus is also worth a mention.

https://www.youtube.com/watch?v=X32dce7_D48

Side note: ABC in Australia have a show called Teenage Boss. About kids using math outside of school.

[u/introjection]

Profrobbob and Professor Leonard helped me from trig all the way up to calc 3 as a slightly older student, can't recommend them enough.