Transcript for:
Lecture on Limits and Functions

we got to start with some Basics is that 1.2 or 2. 1.2 we're skipping the chapter and I I was going we're in trouble now yeah I'd like you to read the rest of chapter one you know just kind of get it and then we'll be on suck some teachers do that I've heard anyway no we're 1.2 my bad I I saw I just like the number two so I thought I'd put it up there on the board see what happens you know we're going to have to start with some Basics some Basics that we know are truths and then we'll be able to figure out some properties use those properties to discover how to compute some limits so first little basic here the limit as X approaches a of a constant C or C is a constant what's the picture what's the graph of a constant show them with your hands with the picture of it sure yeah exactly what is it yeah what's the picture of a constant of of a function equal in a constant yeah it's not this way right it's this way that's a constant it doesn't doesn't change slope wise the slope is zero it just goes straight across true so if I were to draw the picture of this you would agree that that would be my function right well here's the cool thing if I take the limit at any point any point whatsoever so for instance here's my a here's my a I I just picked a random point doesn't matter what it is okay it's arbitrary what's the limit as we go from the left and the right firstly is it the same thing yeah it's not not gapped or anything it's going to the same thing uh what's the limit as we go from the right what's the what's the Y value as we approach the a value c c how about this way so the limit is what that means is the limit of a constant is the constant itself that's cool where C's a constant that's nice right that makes sense graphically we just kind of interpreted so every time we take a limit of just a number that's a constant we just have the constant that makes things nice for us number two how about the limit of X as X approaches a so uh what's the the graph of just X like yal X how's that look line that passes through the origin it goes through the origin great because the slope would be one and the Y intercept would be zero so this is just if you show me with your hands how would it look yeah just like that not like this not like this but a diagonal now I'm going to pick another arbitrary point a the question is if this is yal X just our our diagonal line if you plug in a to y x what do you get out of it yeah you get out a true you plug in two you get out two you plug in three you get out three you plug in four you get out four you with me on that so my question is as we approach as X approaches a from the right and from the left what's the Y approach from the right and from the left a it approaches a so from the right our function the Y value is going towards a true from the left the y- value is going towards a does the limit exist it's the same from left and right and it's equal to how much cool so here's one here's two says the limit this is really interesting it says the limit of a very like X just just a simple function like this is the same thing as if I were to just plug that number in so if if I'm taking the the limit of some function x uh at at some point a the answer is just a that's it that's kind of nice right why because graphically we can see that last one amazingly with with this stuff uh we'll be able to do everything else it's kind of cool this I'm just going to write it out because we just covered it a second ago this is why we covered a second ago how much is the limit of 1 /x as X approaches zero from the left hand side it was on the board just a couple minutes ago how much is that very good negative Infinity how much is the limit of 1x as X approach to Z from the right hand side what was that now last thing we're going to do firstly are you okay with these three basics okay these two this one we already knew these two are kind of interesting they're really unique and we're going to use those to solve some limits here in just a bit to evaluate some limits I should say before we do that though I got to tell you the properties of limits we're going to leave at that for today and I'll show you how to apply these next time so so basic properties we're going to assume that we have two functions and that the limits exist for each one so the limit of f ofx as X approaches a exists the limit of G of x as X approaches a exists notice notice please that these limits are that the x is going approaching the same exact value you see that the a is the same it's got to be the same for this really to work okay so it's got to be the same approach M I just made that word up I don't got to push same same number here's your properties there's basically five things we can do they're all quite interesting first one this is why both of those have to exist is because we're going to do this if you have a limit you're not going to see why this is really interesting until next time so make sure you come back on Friday uh if you have a limit where you have a function inside of it added to another function no matter what that function is you can separate that function so for instance you can do the limit as X approaches a of f ofx plus the limit as X approaches a of G of X that's legal you can separate limits by addition it's kind of neat right by the way this is why you had to have the same letter here because we we have to do that and you can also combine them that's legal to do as well so these are not oneway streets um let's see I'm going to make this a little bit easier for us I'm not write the second one I just need you to know that this works for addition and subtraction that than nice it also works for multiplication you can separate limits by multiplication so that's legal to do you can also separate them by division which means if you have limit as x appr a of FX GX that's the same thing as the limit of f ofx over the limit of G of x that's true but you have to have something what can't this equal you can't have that so the limit of G of X cannot be zero last one the one that I find the most interesting this is really cool I know we're a minute or two over sorry about that uh if you have the limit raised to some power some function raised to a power this is the coolest one you're going to see why in next time but you can actually do this this makes it so you can do a limit of the function and then raise it to the power that's a neat one that means that you can remove an exponent from your function with limits that's very cool uh what's what's fascinating about this is it even works with this because that is an exponent right a radical is an exponent you can remove a radical I'm going to leave it like that uh you can take a radical outside of your limit going go we're going to do many examples of those things but have those written down we'll start with that next time well if you remember from last time what we're talking about is limits and I I think the last thing I did leave you with was the properties of limits and there's a few things that we can do with our limits now now one thing we knew was that from our basic limits our little bases up there the limit of a constant equals that constant that was real nice also the limit of just a simple variable like x equal whatever the X was approaching that that number so those are the two Basics I gave you and then I gave you some properties basically with our properties it says you can split up limits by addition you can split up limits by subtraction you can split up limits by multiplication you can also do it with division as long as the limit of the denominator is not zero you can also this is the coolest one I think anytime you have a function taken to a power inside of a limit you can take a limit of the function and then take the power afterwards so that means you can really alleviate the problem of having an exponent within any limit are you ready to see how we can do that to to evaluate that limit without doing those silly like check a two in the middle and then find out from the left and from the right remember doing those right those tables we don't want to do that anymore we want to use the properties and our basis for limits to be able to solve this thing you ready to do it so here's what we can do if we look at this thing and say okay limit as X approaches two of this function the first thing I know is that I can separate limits by addition and subtraction you with me so what this says is okay well this can become the limit as X approaches 2 of X Cub minus the limit as X approaches 2 of 2x plus the limit as X approaches 2 of s now a lot of people are going to ask do I have to put the limit as X approaches two every time yes yes you have to do it it's like uh it's like asking someone to take the absolute value you have to write absolute value until you AB actually take the absolute value of something so you have to keep on writing limit until you actually take the limit if you don't well you're saying you've taken the limit and that's what it was so it's not a true statement so you have to write the limit as you're doing these problems now let's use those other rules one other rule said that anytime this is the cool one the one rule number property five says anytime you have a function to a power what I can actually do is take the limit of just that function and raise it to the the power later that's true you can do that means I can take the limit of x and then CU whatever that happens to be another thing said that I can separate two limits by multiplication so this would be the limit of two times the limit of x plus that's already a constant we'll deal with in just a how many people are okay with this so far do you see what we've done with these properties we actually broken up a pretty intense polom into limits of constants and limits of just X terms have you seen that what's interesting is I want you to think about this can you do this for any polinomial you can always separate by addition subtraction right you separate term by term then can you always take out exponents yes yes you can can you always separate multiplication absolutely so you could do this with any polinomial you follow me on that one well now check it out since we've broken up into such simple components you can now use those two B there's only two Basics use those two Basics to actually evaluate this limit can you tell me what's the limit of x as X approaches 2 it was one of those Basics that I gave you I drew the the pictures of them it said the limit of x as X approaches a was a it said the limit of C no matter what was C you with me on that so limit of a constant is the constant so limit of x is a whatever it's approaching so you all tell me now what's the limit of x as X approaches two it's just two yeah two and then I can Cube it minus oh how about this one what's the limit of two doesn't even matter what I'm approaching well what's the limit of two as I'm approaching two the limit of a constant is the constant so what's the limit of two two it's two it never changes two times uh how about this one what's the limit of x as X approaches two it is two plus what's the limit of seven no matter what you're approaching what's a limit of seven you with me on the seven it's a constant yeah so we had that those two Basics limit of a constant is simply the constant itself it's a horizontal line the limits always be the same it's just seven can you add that together how much is it 8 - 4 + 7 looks like 11 to me did you get 11 as well that means we've just calculated our very first limit without doing tables and without looking at graphs we're able to do this now can I can I ask you do one thing for me please just check this out can you go ahead and just in your head or off to the side take this number two plug it in there and tell me what you get can you do that for me what's F of two wait say that again 11 did y'all get 11 when you plugged in two wait that's weird is that a coincidence no no it's not a coincidence cuz here's what we can do oh not two sorry 11 it says well if you can separate a polom into its terms and then its individual components raised to powers and multiples like that essentially all you're doing is plugging in that number for those spots for your spots of X right for your variables of X basically you can do that here so here's what I'm going to say you already determined I already asked you this whether this would work for every polinomial you said that it would it or wouldn't it can you do this with every polinomial polom of things that looks like that can you do it for stuff like that absolutely you can always separate them take Powers outside break it up into a constant times a variable that means that to find the limit this an important thing to find the limit of any polom all you have to do is evaluate that function at the point a whatever that is this is going to work for every polom so this idea works for every polinomial is point a always given oh yeah yeah and so it's never it won't be it always be a given in the for initially yes this number is always given to you has to be given works for any polinomial in in plain uh well in mathematics I'll say it in plain English in just a bit here's basically basically what all this stuff says it says that the limit as X approaches a of some polom do you understand the notation the limit of a a polinomial that's a polinomial as X approaches some number is equal to P of a what's P of a say that's exactly right says evaluate the function at that point basically here's what it says in English to find the limit of a polom just plug in the number that's what it says can youall do that oh yeah that's great it says to evaluate the limit of a polinomial plug in the value to find the limit of polinomial just plug in the B which is a in our case you want to see a few more yeah hopefully you're not like nah done get all the hard stuff just leave like this you want the hard stuff don't you yes not really yes limit of x 5th - 3x + 4 all to the 3 power as X approaches 2 that's what we're asking here for now the question is could you could you if if they wanted you to could you do this whole process and do the same thing yeah yeah absolutely you can pull out that that three that power three you see it you can separate all the terms you can separate out that power five you separate the three and the X plug in those numbers in those two spots and it would be essentially just valuation of that so how can we figure out this limit just plug that number why don't you do that on your own plug that number in there substitute that in there because we know that the limit is going to equal whatever this function is with evalu at the point AAL 2 was it 30 to 30 power or something like that M 27 27,000 27,000 how many people got 27,000 20 is it 27,000 I hope so not everyone raise your hand are you guys okay on how we got those numbers I need to know if you're not then I can re explain it but not your head if you're okay on how we got those numbers the 27,000 yes so do you see that what we're what we're actually doing is saying okay I know this limit is the same thing as evaluation when I plug in the number two and then Cube it 32 three yeah I know I do many twos 32 - 6 like 26 + 4 that's 30 30 to the thir power and you get 27,000 you follow me on that one okay and that's exactly what that means hey you just evaluated your first limit don't you feel proud no oh well you're going to you're going to you get a pla sure eat a bunch of candy you get lots of plaque get it get it plaque versus plaque oh it's so whatever I know but the way not all of our limits go to two I'm just using two so you know what about that now is that a polinomial it can the answer is no no that's a rational that's a rational function is it a polinomial but could you still do the same basic thing and answer is yeah you can because of I think it's rule number four or your property number four three or four one of those things which says that you can actually separate your division can't you provided that your denominator is not what will that make the denominator zero will this point make that zero then you're fine then you're fine you can do that so because we can do this whole step this is the limit as X approaches 2 of 4x^2 + 1 over limit as X approaches 2 of x - 3 well now look at it is that a polinomial yes is that a polinomial yeah they're both pols the that's polinomial and that's a polinomial so this says you can evaluate that by plugging in two and you can evaluate that by plugging in two the only thing you got to check for is that the denominator is not zero which we already know right it can't be zero on the denominator that would be a problem so do you need to show me this step no no just plug in the number if it works then you're okay if you have zero on the denominator then you're not okay okay do you get the idea though do you see that we can separate division provided that this number does not evaluate that polinomial make it zero then that's fine so we're going to separate that we don't have to actually show this part it's just I'm showing you once that it is true because now it makes it a polinomial it makes it a polinomial and we know from here hey you can evaluate a limit of a polinomial by just substituting in that number so therefore we can do it here uh have you substituted that in yet so it's going be 4 * 2^ 2 + 1 2 - 3 17 over7 notice how here I have to write the limit here I have to write the limit here I have to write the limit as soon as you evaluate you don't write the limit anymore do you guys see the difference there so yes you must write the limit until you actually evaluate it once you do that you're done no more limits you just have those numbers but you have to write it up until that point question um say that X was approaching three would you still be able to you would glad you ask that question can I answer that in like 30 seconds okay okay cool love those don't you love it like when you're thinking about something and you're going to make your next point because you're a teacher you know that's kind of cool you want to do then someone goes hey what up with this you're like a that's my next Point don't you love that it doesn't happen to you it happens to me all the time I love that uh actually it's going to be a minute and 45 seconds all right do you have the time on that sheet of paper maybe I'm not OCD how about that one is that a polinomial please shake your head no no pols look like this that's it that's pols no roots no denominators that's pooms so where this is definitely not a polinomial but I want you to see what you can do do you see that you could pull the cube root out of it once you did that you could separate by division like we did in this problem so essentially anything will work provided you plug the number in and you don't have any domain issues so basically if you plug this in it doesn't make that zero you're fine it doesn't make well actually that's a cube do even matter cube roots you can plug in any number that's fine so provided that doesn't that's not zero if you plug in one does this denominator go to zero no then you're fine why don't you go ahead and plug in one and see what we have notice I have my limit here we could do all this work right we could separate the cube root we could do that we could show that we could separate my limits by division and show that as well but you just need to know that we can plug in a number into a rational or any type of problem that we have provided that we don't have that denominator zero and don't run any into any domain issues so if we plug in that value of one we get the do we still the cube root yeah don't forget about the cube root how much does that give you well if you're going to say it say it how much it give you cube of six cube root of six cube of six pretty sure you were able to get the cube root of six okay I'm 10 12 seconds over now let me show you this problem firstly do you have any questions on the whole polinomial idea breaking that up any questions on what we're allowed to do over here we're allowed to uh evaluate limits by just substituting numbers provided we don't run into any problems problems would be zero on a denominator or things that don't validate our domain that would be a problem or sorry invalidate our that don't invalidate our I think this is now going to answer your question in every previous problem we were able to just substitute in that number into our limit because we knew we could break it up into uh forms of pols or or basically because they're pols Powers with individual X's multiplied by constants right and we could do those limits that's why we were able to do this stuff is because we could have shown all that breaking it up and just evaluated constant limits and limits that just had x's and then taken the powers later the the only difference with this one is if you try to do that if you separate this by division the numerators could work just fine because you're going to get how much on the numerator that's a polinomial right you get zero is it okay to have zero on a numerator sure yet on a numerator yes now what would happen though if you try to substitute in this two down here zero the limit of your denominator would be zero is that a good thing to have that's a problem we can't ever have that what do we do the top is difference of squares the top is a so wait a second though are you saying that if I do this I know you all know a difference of squares right what's it going to factor as are you saying that I should be able to simplify those out of this expression but wait a second wouldn't we be simplifying out a domain issue you're not actually say that again say it not going to actually ever get to the point two on the gra so are we simplifying out a domain issue not he's exactly right are we actually getting to the point two no so really this is kind of weird but with a limit we're technically not making anything any mistake here we're not actually getting to two we're actually getting that point so when we look at this we go oh okay we could just simplify that out we're not simplifying out a domain problem because none exists because we're not actually getting into two you just can't evaluate the limit by plugging it in because according to this function yes you will at that point that's the difference between pols that don't have issues and things like this that do that have domain problems if you're trying to find the limit of this sky as you're going two well there there's a problem with that there's a hole there right there there's do you recognize firstly that there is a hole and not an ASM toote here what was the difference between a hole and an ASM toote according to the algebra oh you better know you got to know that astes don't factor out say that louder ASM totes don't factor out you're right ASM tootes don't factor out holes do factor out do you remember that so if you can simplify it out that's a hole if you can't you have some sort of an ASM toote so this case yes we have a hole that means that if you have a hole in your graph can you actually find out where the function is at that point you can't even it doesn't even exist so we have to do limits to find that out but in order to to do that we do have to simplify this this function we can Factor we can simplify because we're actually not getting to the two this is okay it's okay because x x never equals two never gets there really close yes so what we can do now is say all right then this according to the limit is exactly the same thing as that is that now a polinomial sure can you plug in two how much you going to get four is four that's kind of neat right oh this should be amazing to you be like yeah we can simplify this stuff out because we're not actually getting there there therefore we change these weird looking uh holy functions not because they happen on Sundays but because you know we have and then we can just plug in the number because now it's a polinomial and we're good to go so if you can factor and simplify then you can find the limit does that make sense to you that's awesome few more okay we're going to make them a little bit more advanced a little bit more advanced Till by the end of the day we'll have some pretty unique things going on okay now before you get all crazy and start factoring everything that you see you might want to just check that point and see if it's even a problem all right because sometimes they might give you that stuff and there's no issue it'd be like this example this would have an issue if we were going to the limit of x equals or X is approaching three wouldn't that be a problem but we're not we're going to two so there was no problem here so at least check and make sure you actually have a problem before you start factoring things out make sense so try it do I have a problem if I try to evaluate at4 yes yeah I do I get Z over zero right that means you should be able to factor hey if you get Z over Z I told you this was the property of mathematics if you get 0 over Z when plugg in the same number that means x minus that number is a root that means somewhere over here I can guarantee you're going to have an X+ 4 cuz x - -4 is x + 4 this will be x + 4 * something this will be x + 4 * something and then we should be able to simplify out that problem why don't you try that right now factor that I want you to simplify it and then see if you can evaluate the limit of what you get out of that should be your quoti I suppose did you factor was there an X plus 4 like I magically predicted I love when that happens too so if you were to factor this out by the way have you noticed that when you're factoring you still must have the limit notation you see that on the board you've got to have limit look at here's a limit here's a limit here's a limit when did I stop using limit when I finally actually was able to plug in the number do you see that you must have a limit I will Mark you points if you don't have those limits if you go I'm going to write this limit but not that one not that one I'm just going to my answer mm that's not how limits work if you say you don't have a limit here you start Crossing stuff out and then you magically plug in you've made domain issue if you're with limits that's okay so you have to say I'm working with limits here until you get down to this point do you see the point all right so we still have a limit by the way that's going to be annoying I know because you're going to have lots of steps on some of these limits but you need to do it we're going to factor we're going to factor see anything that simplifies out of our problem now some of you might be asking well wait a second don't I still have a domain issue because if I plug in three I have a problem don't I still have a domain issue the answer is well yeah sure at three but are we trying to figure out what's happening around three I don't care about three what point am I trying to figure around yeah so I've gotten rid of the problem cuz as soon as I cross out this x + 4 now when I substitute in4 it's no longer an issue can you now evaluate at4 let's go ahead and do that notice how I've written limit limit factored limit still and now I can put well this is going to equal 2 -4 - 3 I've now evaluated my limit I don't write the limit notation any longer because I I don't have any more variables as soon as you substitute it for your variable that is your limit we get -27 as our how many people got 27 feel okay with this is this understandable you guys starting to get the the concept of working with this stuff it's kind of interesting isn't it what we can do I hope it's interesting to you oh how you have it up there cool all right oh my gosh what are you going to do first do you think but before you factor don't get crazy on factoring what are you going to do check you're going to check it yeah check the five because if it works well there's no reason to factory that would just be a waste of time probably confuse yourself too like Wait nothing simplifies oh no now if you try the five here do you have a problem okay so if you do have a problem then what do you do now you factor so if you haven't factored already go ahead and Factor what's the top one factores plus 2 - 5 how about the denominator what do we got there ohus 5us 5 all right did you all Factor it correctly all right thank God good at least we factored it correctly do you see anything that simplifies wait not not both x 5 one so we go here and here yes and what we get out of this this is important example for you to see you get x + 2 x is still approaching 5 x - 5 try to plug in the five what are you going to get you going seven over zero is that still a problem oh man so wait a second what do you do if you still have a domain issue on this thing I've now given you two cases I've us in cases where the problem the domain issue is a hole do you see that it's a hole and you can get rid of a hole it's a removal discontinuity and you can find the limit of that it's very easy you factor it and then you evaluate now I've also given you another case where not only do we have a hole here we actually have what is that when you can't simplify out the domain issue that is a vertical ASM toote so now we have this issue what are we going to deal how are we going to do that well this is where you're going to use what's called a sign analysis test listen you you really do have to comprehend that if you can get a z over zero like you did over here you're going to be able to at least Factor it and simplify something right sometimes as I've shown you it will get rid of the problem area and you can just substitute in a number and be just fine that happened here and that happened here now' your have be okay with that one now other times sure we got 0 over zero we factored it we got rid of a what looked like a problem area and it was that's like a hole however can you get rid of this x - 5 in any way whatsoever if you can't what was that called again it's ASM toote what you have to sign analysis test here's what you do with a sign analysis test you put the numbers that will make the numerator and the denominator equal to zero put that on a number line what numbers am I talking about here five definitely five because we want to find out what's happened at five and that was our problem area so five is going to have to go over here here somewhere so far so good now what else do we need now ne-2 is interesting we don't really need it cuz really we're looking at this area however we need to know what happens between this range you see what you're going to do is called a sign analysis test and that means you're going to find the sign in this interval and the sign in this interval the only reason why we need this number is to make sure that you pick something in this range because if you don't if you pick something in this range this is where the signs change from positive to negative does that make sense to you so you have to find something in this little range and this range over here it could be anything over here but in this range it's got to be between these two numbers that's why we put the -2 we're not taking a limit to2 we don't care really we don't care we want to find out what happens at the five is that clear for you are you talking about like the trick function sign no no this is easier okay there's no trick function here okay function I'm going okay see get there where's these Tri functions coming from I hate calculus no no trick functions this is basic basic okay this is just plugging in some numbers sign as in I am signing okay sign like like a sign like sign oh I get it sign like sin we're not sinning we're signing here we want the S let me WR that sign analysis analysis is another one of those things you can't really abbreviate that well anal got it the sign analysis that'll go viral on YouTube that' be awesome so again the reason why you have to have the negative -2 is because your sign will change in these two intervals and we don't care what happens here but we want to make sure we pick the correct number for this interval we don't want to be picking ne3 because it's going to give us the wrong interpretation of what happens around this now you have to know what happens around the five and you already said it once or twice before is this a whole or an ASM toote as it's an ASM toote because I can't get rid of it you follow so you know for a fact it looks like this you know for a fact that you you have an ASM toote right here right so you're going to be in one of four situations watch carefully you you are either going to be going like this true or like this or like this or like that it's one of those four situations and the sign analysis will tell you what it is here's how you do it I want you to take a number in this interval what's a good number in this interval pick six you can't pick five you well obviously you can't pick five pick six plug in six and tell me what you get plug it in not the original but no no you know what what you've done here is you just simplified out a whole it's going to give you the same exact answer so plug it in here what' you get you got eight CU you did eight wait what' you get oh right8 over one one so eight over one is positive 8 that's what I'm looking for I really don't care about the value I care about positive or negative what you get positive or negative make sure you know how to deal with your signs okay so you you got a positive you know what that means if you got a positive are you going to be going down to negative infinity or up to positive Infinity definitely positive Infinity because you're doing positives here you know it's an ASM toote already somehow you're going to be looking like that you okay with that so far okay now try another number in this interval again the reason why we have aa2 is to stop us from going too far so what's a number in here pick a better number than four pick zero zero is a great number interval so plug in zero what' you get 2 so is it positive or negative am I going to be going up to meet that other ASM toote at Infinity I'm going to be going I don't really care how the graph looks honestly I don't I just really care what the ASM tootes are what what the function is doing around the ASM toote because that's the limit is what's it doing around the number do I need to plug in a number over here pass -2 no this is just to stop me so so you got to tell me does the limit exist cuz we're talking about five does the limit exist at five or not no if it did they'd both be going up or they both be going down okay if they're like this then it doesn't exist but this is how you show that it doesn't exist you don't just automatically go I can't do it does not exist okay because you could be wrong all right it happened before it does not exist because you showed it with your sign analysis test so on your test when I give you a problem very much like this which I will do and I ask you to show your work or I might even say show a sign analysis that's what I'm talking about I need you to prove to me prove to me that that limit does not exist so if they say show the limit doesn't exist that's it okay if the limit does exist you'll be able to simplify out the holes if it if it doesn't exist well then you have to show a sign analysis test show that you have an ASM toote and it's going like that or some other way that you you cannot make those things go together how many people understood that what we just talking about good all right couple notes for you so I'll write this out so you can remember later first one if you get 0 over Z what that means is you have a common factor at that point X Plus x minus that point so if you get 0 over 0 what you're going to try to do is factor and simplify factor and simplify 90% of the time that's going to work just fine that's these examples over here all right the other percent of the time let's say you factored you simplified you still have an issue so that's number two if you cannot simplify the problem okay if you can't get rid of the problem if you can't we'll say it this way if you can't cancel the problem because youall love the word cancel right if you can't cancel the problem that means it's not a hole if you can cancel the problem it's a hole if you can't cancel a problem that means you have an ASM toote at that point what you're going to need to do is evaluate with a sign analysis so two two situations holes cancel them great done not holes those are ASM tootes sign analysis will show you what that does around the ASM toote if they're go into the same thing it exists positive netive Infinity if it's not then it doesn't exist and that's the only way you can show that for us so if you cannot cancel the problem on the denominator check with sign analysis because the limit might not exist because a limit might not exist how many people feel okay with what we talked about so far okay I'm going to show you a couple more very unique examples we'll talk about those um things you you really have to know exactly what to do uh when you see them so I to make sure you see them I've started writing in all caps thanks to you you're welcome my handwriting is so atrocious that means bad so condescending when I just did you that's horrible um sorry um it was really bad until I started writing all all caps and now I have trouble even writing limit like like that also my dad was a carpenter and a a draftsman and so they used he used all caps all the time so I want to be like him so I started doing that too he's much better at it all right now we got this this situation here limit x -1 over square root of x oh my gosh well the first thing you try to do is what try to plug it in do you have an issue now you have Z over zero true unfortunately for us these aren't easy to factor right you do you probably do have some sort of factor out of that but it's not easy to do polom are easy to factor that's fine but with this that that theorem really doesn't hold that much water because it's not easy to factor that so okay what are we going to do any ideas conjugate have you ever heard of the conjugate the conjugate is those two terms with a different sign in the middle if we do that that's one way you learn in your intermediate re course on how to rationalize a denominator or rationalize a numerator or basically just rationalize out a square root so if we do that that might help us so don't forget the algebra that you know sometimes we can rationalize if we do we'll rationalize < TK X - 1 oops sorry plus one over > x + 1 you have to use the conjugate though other if if you don't well then you're going to just make your problem worse because you're not going to get rid of a square root it's going to be within a middle term of your problem so you have to alternate those signs you guys okay with that you sure so use that conjugate now let's see of course we have parenthesis here now I'm going to give you little piece of advice generally generally you don't want to distribute in this case the numerator if you're a rationalizer the numerator you wouldn't want to distribute the denominator because ultimately you're trying to simplify out something are you with me on that you're trying to simplify that out so I'm not going to I'm purposely not going to distribute the numerator unless I absolutely have to if I run into an issue so right now I'm going to leave this as x -1 and theun of x + 1 why because I'm trying to simplify stuff that's why now on the denominator you tell me uh when I what do I have to do with this I do have to foil or or distribute so that means every ter times every term U tell me what's < TK of x * X uhuh now do you see what happens and why we use the conjugate if you distribute this we get X we get plus < tkx we get minus rootx what's going to happen with this that that's why we use the conjugate and then last you're going to get what you get xus one do you see why we don't distribute the numerator if you distribute the numerator you got to you got to mess of crap up there right you're going have to refactor it's not going to be easy if you don't distribute it the factoring is obvious the simplification is obvious what are you going to simplify well at least I hope it's obvious is it obvious to you what you're going to cross out all right good wo yeah kind of obvious all right well our limit as X approaches one notice I'm still writing limit is the what what what do we have left are you going to have an issue oops that doesn't look like an X are you going to have an issue if you plug in one it's not negative it's positive one so we're okay we have no denominator anymore because we rationalized it it went away because we were able to simplify it if we plug in one what are you going to get that's your limit how many people feel okay with this so far would you like to try one on your own let's do that I'm going to give you a little bit more complicated of one not too bad and if you get stuck on it no big deal that's fine but I want you to at least think about it while you're here this will be where we end today so limit as X approaches Z of < TK 1 + x - 1/ X are you going to have an issue here yeah I mean straight up we're going to zero and that's over zero that's a problem so what could you possibly do run away yeah done no we're not going to just leave the problem we can't plug in zero the only thing you can do do here is what do you think can we use that same idea but only in reverse let's try that go ahead and do that multiply by the conjugate conjugates have to have different signs they have to that's got to be the same thing on the numerator denominator it has to otherwise you're not multiplying by one and if you're not multiplying by one you're changing the problem you can't change the problem also one more thing I need you to look up here at the board when you do this that sign doesn't change it's only the thing after the square root so this stays the same did you all multiply by exactly that do you see how that is the conjugate we have the square root whatever that sign is and that whatever that constant is that's what we have here the square root the different sign same constant same exact thing why you why you need the same exact thing in case you're wondering well Mr Leonard why don't you change that sign as well what you're trying to do is multiply this in such a way that you actually eliminate the root so when you multiply this one times this one the whole entire root goes away right the only time you can do that is if the roots are identical so you can't have different signs otherwise they're not identical hey which one aren't we going to distribute here the numerator or the denominator denominator don't distribute the denominator by the way I'm saving your lives here if you distribute the denominator you have to factor it again that waste time so literally I'm saving your time there go your life nice right I know I'm such a nice guy today so on the denominator I know I'm going to have X and then the < TK of 1 + x + 1 I'm not going to distribute that the numerator yes you're going to distribute the thing you're trying to rationalize so if you were to distribute this once youall help me along here when I distribute this what's the first expression I'm going to get 1 + x very good okay and then what so I'm going to getun x 1 + x positive < TK 1 + x negative that's going to be gone that's what the conjugate does for you and lastly I'm going to get so this is 1 + x - 1 anything else we can do with that like term anything else yeah combine like terms what do you get if you combine like terms on the numerator that's kind of nice cuz one and negative one and this is why we didn't distribute the denominator because if you look at that that's what we're trying to simplify out right trying to get rid of that 99% of the time this works out for you I wrote it back it doesn't matter now that you see the X and X as a factor on the denominator those things are gone what's on your numerator please oh zero is not on your numerator what's on your numerator come on now people one one yeah when you cross something out you don't get a zero you're actually factoring that out saying X overx is one so you have a one up there uh by the way please don't make the the intermediate algebra mistake of doing this a lot of people do this when they're just beginning they go oh yeah I cross everything out therefore I have that is that true no no you don't have that you actually have this if you forget that what you're going to end up is end up with is the reciprocal of the answer that you actually want that's not good hey now can you plug in zero and be okay yeah even though you still have X on the denominator look what happens what's 1+ 0 one sare root of one plus two said two I was one step ahead of myself I'll just write on the board there you go one2 would your ra have feel okay with our our limits good deal now you know we went a long way today we now know how to evaluate and compute these limits uh next time I'll show you how to do some peace wise limits I'll show you some trigonometric limits and then we're off to a fun start did you have fun today you have fun today all right so welcome back we're talking about limits uh we're going to start talking about peace wise limits now now for us we found out that when we're taking a limit usually we can cancel out a problem or if we can't we'll use a sign analysis test because we have an ASM toote hopefully you practice that in your homework were you okay with that idea so generally what we're trying to do is plug in the number plug in the where X is approaching if it works great that's your limit if it doesn't work well then you have to factor do something to find that limit typically factoring we'll be able to simplify that a little bit or we found out we can rationalize denominators and numerators if we have square roots we do things like that to work around that limit now what we're going to talk about today is some some different aspects to that we're going to talk about peie wise limits I'll show you what you can do with those things it's going to be kind of nice just go follow me on this then we'll talk about trigonometric limits and that'll that'll end our day we have a lot to talk about on trigonometric limits so piecewise limits here's basically the idea when you're talking about pie wise limits and peie wise of course means different functions all matched together that have different directions for each part of it you have seen PE wise functions before PE wise limit says what's the limit as we're approaching that interchange basically does it exist does it not exist what what what does it happen to be so our idea is we're going to have to take some one-sided limits because each piece is different right we're going to take one-sided limits and see if they match up if they match up great limit exists if they don't match up then no the limit doesn't exist are you with me on this I'll give you a real nice way to do this a graphic organizer hopefully this will help you out and then of course if there's questions man ask so let's let's talk about this our idea is we're going to find some one-sided limits and we're going to see if they are equal let's start off with an example let's say that I give you this I say that your function is actually made up of three parts the first part says you're going to do 1/ x + 2 if x is less than2 I say you have a different range you have x^2 - 5 if we're between -2 and pos3 and lastly our last little step we're going to be the < TK of x + 13 if x is greater than 3 now because we have a peie wise limit we basically have in this case three different functions that means we can't just look at this thing Hammer at it one function and find out what is the limit uh of of any particular place what I'm going to ask you for is can we find the limit at two and three as X approaches two and three why two and three well if you look up here that's the only place that could po possibly have an issue right is -2 that would have an issue whether everywhere else it's continuous the limit's going to exist no problem same thing here that's continuous everywhere only problem would be maybe at the end points of two and three here the only problem could be at three with that function so we're going to talk about the limits as X approaches -2 and X approaches pos3 are you with me on that so that that's our idea here what we're going to have to do is find our left side limit for each function and a right side limit and see if they meet up somewhere if they meet up at those points now here's the way that I like to do this first thing I like to do is draw a representation of your of your graph of your number line basically just like that break it up into the places where your pie wise functions broken up what are those key points what numbers delineates our function from the next piece of the function where does one function start and one function stop basically two that's a key point right so we're going to have to have -2 on here somewhere what's another key point for us let's call this one function one and function two and function three can you tell me what function is in this range over here what what function takes over for this interval function three because it says for any X is bigger than three I'm Now function three does that make sense to you so I know that I'm looking at function three here what interval or sorry which function takes over in this interval function two for sure and that leaves this one with F1 let's make sure X is less than -2 function one takes over do you understand that this is how actually our graph should look should be function one then function two then function three and I'll draw this graph for you at the end uh just so you see what this really does look like you with me still so far though so here's the idea if we want to find the limit as X approaches -2 and the limit as X approaches -3 we don't have to find one side limits for all three functions just the two functions that are approaching that number number so for instance if I want to find the limit as X approaches -2 if I'm going from the left if I'm going from the left which function am I going to use function what from the left I'd be using function one does that make sense to you now from the right as we're approaching -2 which function am I using from the right function two yeah that's right so so around this function I'm going to have a one side limit from the left using F of one uh sorry the first function then I'm GNA have a one side limit from the right using the second function does that make sense to you it's a way to picture this how about around three we want X to approach three how about you guys over here what function are we going to use as we approach three from the left function function two that's right that's in the interval how about from the right function function three we're not using function one over here are we we're not using function three over here we're just looking around that that number what functions we have so now because we have this gives you a pretty good idea of what your graph looks like right this is how you make up your one-sided limits now okay so I'm going to take a limit as X approaches -2 from the left a limit as X approaches -2 from the right and I'm going to see if those two things are equal if those are equal then we'll have a limit as X approaches -2 in if not well then we won't do you all understand the idea so far are there any questions so far on the idea okay so let's go ahead let's try this um what's the specific function I'm going to use for this one this is -2 from the left hand side you all said that was function which one function what's function one one now write that this one by the way is going to be the hardest rest reason how about going from -2 from the right hand side what function would we be using isn't this kind of nice you just look at that can't you kind of cool look at that one uh Now function two what's the actual function two so write that so we said for this range from the left Function One this range from the right function two we're going to see what these are if they're the same thing then our limit will exist if they're not the same thing will our limit exist okay so we have that idea down now how about we set the other ones as well let's start talking about the as we approach three so we're going to need again a limit as X approaches three from the left three from the right if they are the same we'll have a limit at three let's fill out the functions though let's fill up the functions as we approach everybody three from the left hand side which function are we using and function two again you said that was x^2 - 5 so we'll write that how about from the right hand side what function are we using function three what's function three I'm assuming your mumbling was sare < TK of x + 13 was that what you were mumbling I 13 okay now here's the deal I'm going to work this side this these ones first then this one cuz that one's a little bit more difficult let me pait something to you what I'm going to say is that if a limit exists a one-sided limit definitely exists would you agree to that if a limit overall exists then a one-sided limit will certainly exist agreed so a limit existing is stronger than a one-sided limit existing does that make sense to you well here's the cool part about that then and this a lot of you were asking how do I find one side limits check it out if you know for sure that a limit exists then the one side limit will exist plugin three ignore the from the left plug in three what do you get you get four guess what that's your limit why because does the limit of three sorry does the limit as X approaches three exist for that function absolutely without equivocation because that right there's a polinomial right and you know with polinomial is you can just plug in a number unless you have a problem well if the limit at three exists the limit from the left certainly has to it must that's a weaker statement plug in three you're going to get four does that make sense to you let's try it next again if a limit exists a one side limit will also exist at the same number so ignore the the Plus for a second can you plug in three without a problem there what are you going to get you're going to get four saying the limit exists therefore the limit from the right well it has to exist if the limit exists the right and left side limit for sure exists they have to go to that same number that's going to be four now you're able to answer your wasn't that nice and easy just plug them in that's it just plug them in if you can just plug them in nice and easy does this limit exist why yeah I don't care that they're not the same function they go to the same value limit is four ra you have you're okay with that one so far kind of nice right one side limits aren't so bad P wise functions they can get kind of messy we find that over here but not too too bad do you have any questions on this because I'm going to erase it because I need the room any questions on that okay so again if a limit exists a one- side limit will exist find out your one- side limits if they're the same then your limit at that point for sure you got it now let's start over here can I plug this into that function yeah absolutely that's a polinomial so ignore the the right plug in -2 how much are you going to get one absolutely negative one there's the issue okay here's the issue uh ignore from the left can I plug in -2 aha so what you're saying is that the limit at -2 you don't know whether that exists or not right you don't know what's that mean can you can can you cross out any part of this can you factor and cross it out okay so think back what do you use if you can't cross out a problem what do you use the thing you couldn't abbrev that's right the sign analysis thing you have to use that something you can abbreviate it it's just going to look funny in your paper but you have to use a sign analysis here so if you have a sign analysis at -2 what we care about what's Happening Here what is happening at -2 do you have a hole do you have an ASM toote which one folks you all should be a telling that you all should know that at this point if cannot cross out your problem what is it it's an ASM toote so what do you have as you approach -2 it is certainly an ASM toote yes it's we just want to know is it going upwards or is it going downwards what could you do to find that out any value any value to the left over here so plug in -3 if you plug in3 to this function are you getting a positive or A negative negative definitely negative so am I going upwards or downwards like that right so here's what you're just finding out what does this limit I don't care from the right look why don't I care about this why don't I care is this the same function on this side of that -2 is that the same function over there no no that's we've already taken care of that one I don't care about that I just care about this so what is this limit equal can you tell me yeah as we go towards -2 from the left hand side it is going like that that's going to towards where so let me recap just a little little bit what you're doing you're breaking up your interval you have three different functions in this case you set up your limits around those breakoff for your your intervals you use left side limits you use right side limits and you see if they're the same some of them are going to be easy like what I showed you over here others of them you might have to do a little bit of work these ones you can just plug in that one you can just plug in this one if you can't cross anything out and eliminate the problem you've got to use a sign analysis test don't get stuck on that okay you're going to have something like this on your test don't get stuck on what do I do oh no I can't plug in -2 what now sign analysis test if you don't know what to do you don't got to throw things jeez my goodness you're in luck it surved you just got that yeah now you just got that with some nice marks on it congratulations don't just get a new motorcycle okay anyway uh so this one yes you plug this in this one well you can't cross anything out you have to do a S analysis test do what you know don't get stuck on it if you can't cross anything out sign analysis that's that's the only two things you got okay so we have this we were able to plug in number for the left of our interval that's the side we were wanting went to negative Infinity do does the limit exist for this this is1 that's netive Infinity last time I checked those things aren't exactly the same so this does not exist but that folks is how you check you don't check by plugging in numbers okay randomly you check by showing your work like this this is showing your work does that make sense to you I don't want little tables or anything like that see look the numbers are not the same I don't care I want you to show me what this is now the way this graph looks if you want to see why this doesn't exist uh what this thing is is a parabola part of Parabola that's symmetrical around the y- axis and intersects it's going from -2 non- inclusively to three inclusively so if you plug in -2 what we get 1 looks like that and three if we plug in three uh we get out of that four that's not really accurate that part can't cross three and end at three but whatever then from there on out we have the square root of X+ 13 which starts at four and takes off something like that and then we have this one which at this point I'm sorry this one this one over x + 2 if you graph that that's some sort of descending function that goes like that at that point does the limit exists as we approach -2 no this is a rough sketch by the way does the limit exist two absolutely because that was that was closed off by the way it's a function because that's not equal to and that is that's okay we don't fail that part uh this part would have had the open c circle around it but it's filled in by that point so we have this Parabola limit exists there absolutely limit exists here no netive infinity and a value Rich have you're able to follow that feel okay with it good deal are you ready for some trigonometry no I know it's supposed to be trigonometry Tuesday but we're we're day early so it's all good was that your drinking party idea drinking party IDE I don't have drinking parties koola limits of treat functions okay there's a couple things you have to buy in on for me to do this properly and the first thing you got to buy in on is that s and cosine are continuous everywhere continuous means you can draw it without lifting your pencil off the paper can you draw s and cosine such that they are continuous do you ever lift your pencil off the paper no so they are continuous so first thing this is going to come up later in our class but I want to say it now s and cosine are continuous everywhere tra tra so because we can do that because there are no problems we can use something that we use for pols you see pols were continuous everywhere if you think about any polinomial I'm not talking about a rational function okay I'm not talking about denominators I'm not talking about roots that we have over here this is a rational function clearly it's not continuous everywhere -2 fails this is not not a continuous ever because it doesn't even exist for part of it okay so we're not talking about that what we're talking about is continuous functions like this one that's continuous everywhere you're not going to have a problem no holes no ASM tootes agreed s and cosine we just said are the same thing therefore if we have S and cosine are continuous everywhere we can apply the same logic and say then the limit of s of X as X approaches a is what do you think it's not a it's not a you wouldn't say this you wouldn't say the limit as X approaches 5 of x^2 - 5 is 5 that's not true what would you do to find it plug in you would plug it in so to find the limit of this what what we do we plugged in right it wasn't -2 it's what we got after we plugged it in so it's not a what is it a s of a absolutely likewise because cosine is also continuous everywhere we can do the same thing oops yeah that's true it says that you don't nothing bad happens basically that would be something bad happening there we go that looks a little bit better what about tangent tangent so my question is is tangent continuous everywhere draw tangent with your hands how's it go look like you're dancing yeah you doing I should have used that I went out dancing the other night I should have used my my SC curves it would have been the hit of the party probably not maybe probably not okay so we got to do a little bit more work since it's not continuous everywhere we're going to have to break this down a little bit so the limit of tan X as X approaches a let's figure this out tell me what is tangent use an identity what's tangent sure so we know for a fact that the limit of tangent is the same thing as the limit of s over cosine true very true very true now there was one property of limit that said you can separate limits by division remember that property kind of cool let's do it okay okay so limit of s limit of cosine do we now know what the ignore the bottom do we know what the limit of s x is as X approaches a sin a sure so this says okay I've got sin a what's the limit of cosine as X approaches a what's s over cosine Aha and you go yay what's the one thing I've screwed up on can't be Z say say that again can't be zero why can't cosine be zero that was the one thing that one little condition about that property if you remember it said you can do that provided that your denominator when you split that up doesn't ever equal zero and so we'd say sure the limit the limit of tan X as X approaches a is T A except at certain points at certain points and that's where cosine would equal zero uh that's specifically is the reason why you get all those Asm tootes on your tangent line is because tangent is s over cosine so naturally the sign over cosine and cos equals z you have an ASM toote remember the ASM toote idea if you can't cross it out you get an ASM toote right that works all time so that's why we get those ASM tootes going on and those ASM tootes occur at X cannot equal or we will have a a discontinuity uh because right here oh let me draw some purple so you see it from the different cosine X cannot equal zero it does you have a problem where that happens is uh every plus and minus < / 2 so piun / 2 3K 2 things like that X or a or either one cosine X can't equal zero at those points you could say x can't equal these things so or cosine cosine a really doesn't matter say cos a a little more sense cosine a can't equal zero does you get a problem so tangent would even be continuous at at these points um plus orus pi/ 2 because when you put those into cosine you get zero plus or minus 3 Pi 2 and so on and you can really you can see this in the domain of tangent if you look at the domain of tangent that's again that's why you get those ASM toes every pi/ 2 remember tangent power two astop 3 power two asmt and so on and so on and so on and negatively as well you guys all right with this so far this is our Basics you guys okay with our Basics start ramping it up a little bit you're like no no this is fine let's just stick with this good we need to practice this a lot too bad said disappoint a lot in this car there you go that looks messy enough didn't it how about that yes sweet do this okay uh what in the world let's just give up call it a day you ready out screw this stuff math we just watch football can't do math and football they're oxymorons mathematical football oons brain injuries and football that goes together not math work okay what can you do with this well I'll tell you something uh you can this is interesting but if a function is continuous you can treat it like a composition is co sign continuous everywhere so what you can do with this is kind of cool what you can do is say all right s cosine continuous everywhere you can do a composition cosine is continuous so by composition actually reverse composition look what you can do you can say instead of taking a limit of cosine of something you can actually do this it's very much like removing an exponent from your limit you can say I don't want to deal with that I want to deal with cosine of my limit that's legal why would you rather deal with that have you seen things like this before yeah actually I think I've given you that exact same example so far in this class you've seen things like that can you do this limit so as long as you can do this limit and don't forget about your cosine you'll be fine it's interesting to think that the angle now is a limit that's weird right so cosine of what are you going to do with that limit what are you going to do first you're going to try to before you factor you're going to try to try to plug it in I mean don't may as well just try that if you plug in one though you're going to get 0 over 0 what does 0 over 0 tell you is factoring going to work or not yes 0 over Z tells you that so you're going to factor it probably going to work now we've seen one case where it doesn't where it was a double root okay that that happens but factoring is something you should try first so we'll have the limit X goes to one of well I know youall can Factor we got x + 1 we got x - one what can you do now sure what was that discontinuity that we just talked about was that a hole or an ASM toote in this particular case hole ohle sure because you can cross it out and there's no more problem left that was a hole now um other question let's see oh I had one more was on my mind ah forgot it cross it out do you have any problems left notice how we have to keep writing limit until we actually evaluate the limit that's important part of it what can you do can you plug in the number and be okay so then what we have right here is cosine of I know I can just plug in the one 1 + 1 what's cosine of 1 + 1 cosine of two done cosine two we can do one plus one I like one plus one how do you feel okay with this one so we can treat that like a composition no we just insert the one with the initial problem why do we have to pull the cosine out can you just let keep writing it the same way and come to the exact same answer yeah would that be still an acceptable way of writing it I don't know okay just showing you you can do it cuz a lot of people go what do I do with that you go well can you still Factor it can you still cross stuff out and the answer is well technically no because it's inside of cosine but as soon as I move the cosine outside see what we're dealing with now is a limit and now it's not a domain problem because you're not actually approaching the one so yeah then you can does that make sense okay so moving the cosine out is just a matter of semantics basically yes yes but important semantics nonetheless so can you do it yes you can do it does that answer your question all right would you like to travel more answer is yes most of the time yeah unless I'm really tired never came down one could always wish I don't even drink coffee oh man if I drink coffee you guys would be in trouble if I drink coffee that day it's like crazy yeah so basically do you actually have to pull out the cosine yeah you can you can that's the reason why this is allowed to do it do you have to show me that I don't really care um it what it basically says is if you can plug in the number you can do it that's that's what cosine being continuous everywhere says to do also says that you can still cross stuff out as you're going through your problem because it's a composition it's continuous everywhere you can pull that out does that make sense to you so can we try to plug in a number here is that okay sure as long as we don't run into an issue IE you you plug in something that's not even in the domain like a negative root or you have a denominator that's zero that would be the only issues you have do you have an issue if you plug in pi over2 here no actually if you look at this that's a polinomial right plus a function that's continuous everywhere that means you could split that up by addition we knew limit rules for that you could do this one by itself just plug in Pi / 2 you can plug in pi2 to cosine because it's continuous everywhere not a problem so that says that just like other limits that we're dealing with try plug it in the problem first if you can no problem so that would be 3 Pi / 2^ 2 plus cosine pi/ 2 uh what's all this going to give you not as a decimal oh come on what's pi over 2^ 2 you can say pi^ squ it's okay Pi s over 4 very good so this is going to give you 3 pi^ 2 over four are you following me on that okay what's that going to give you don't say that loud I want youall to think about it you need to know that I said don't say loud and hold it in zero notice how at this point I don't have to write limit anymore I actually inserted the pi/ two you guys see that that's your answer so once you once you're no longer dealing with variables you don't have to keep writing limits as soon as you evaluate your limit at at that point no more limits but what if you keep doing step after step you know because like I I write it out I do this part of the math and this part of the math and I keep continuing on but once I put in once there's no longer variables then once you evaluate the limit normal once you are able to plug in that number no more limits you've done the limit right yeah over and over and over again well you should you need to write it I had to write it here I had to write it here here but not here because I was able to plug in one now what if we say we write it too much are we on it there's no more limit here so if we wrote limit in there the whole thing is you got to know what a limit actually is right you got to know that a limit is what does the function do as you're get into that number you got to know that as you're getting to that number it's PO number consider everywhere you can actually plug it in and that is your limit okay you say this is still a limit of something no it's not that is the answer to your limit question but it's that was a good question though I'm glad you brought that up because maybe other people were confused about that that's a good question remember for something a little bit more interesting you know what interesting means m mathematics right harder harder scarier scarier yeah it's a little bit scary I'm going to show you something it's kind of cool it is not my Darth Vader mask was pretty cool no I'm lying my parents once told me that I had a stormtrooper mask but I've never been able to find it so I know they're lying they like oh yeah we bought that for you it's it's like when you're like growing up and you ask your parents have I ever been to Disneyland like yeah but really it was just a park down the street with some guy named Greg same thing that's Mickey Mouse no that's Uncle Greg okay okay let's try that let's try that oh okay let's go through the process of finding limit right right the first thing you try to do is what yeah you try to plug it in so you plug it in what's s of Zero no sign of Z is not one sign Z is zero over how much can you factor sign you ever been able to factor sign in your life no no no so can you factor out anything and cross anything out can I just cross out the x and x and get sin yay it doesn't work okay sign has to have something next to so if you do that you have sin in mathematics that's not acceptable you can't do that so we're basically stuck there's no way to do this Pro unless you want to just plug in endless numbers there's no really way to do this but we're going to find a way to do it you ready to find a way to do it some trigonometry here's our idea I'll see if I remember how to do this in a while first thing we're going to do oh yeah we're going to bound this we're going to make up a triangle so let's here what I'm going to do is take part of my kind of a unit circle not really make it better that's better I'm going to take my unit circle what I'm going to do is take an arbitrary angle I can't be specific because of course you want this to work all the time so I'm going to draw that now what I'm going to do from this I'm going to make up two triangles I'm going to drop a perpendicular right from here I'm also going to drop a perpendicular it goes like that okay so let me let me Define a couple things firstly we're going to call X our angle second thing what do I need I need this I need that now what I'm going to tell you is if we if we need to solve for that let's see this is going to be our t x how far is this distance right here from here here to here why so would you agree that stick with me here folks are you okay that X is our angle forly I'm I'm calling this length y Okie doy and and this this is one from from here to here that's one give me the relation sh of tan X then s over cosine yes in the specific instance with a large triangle tangent is opposite over adjacent true what's the opposite y over one one because that is the unit circle well interesting then y = Tan x so this equals Tan x oh let's see one more thing we also need this distance I we need that distance right there so let's say what do we want to do this coordinate is what's the coordinate on a unit circle well that would be for specific angle what's the what is is it cosine or cosine jeez I'm asking you come on you got to know this agree okay yes sinx is typically y for inside Triangle what that what that means is I I'll try not to screw this up too much for for that means that our our X for this inside Triangle is cosine X do you agree that means that our our Y is actually sinx right now also what we're going to be doing is looking at this this not this well this one gives our height but also this one that triangle right there what we're going to be doing is comparing the areas of these triangles so let let me recap just in in 30 30 seconds what we've done we've gone ahead we made up two actually three triangles this one just gave us the height that gave us the height this would be cosine but more importantly this one this height is sinx do you all agree with that one furthermore if I call this y y is equal to tan X because of the relationship of X and one you guys okay with that one okay now we're ready to compare the triangle so we got we've actually have three things going on I need you see all three things first we have a big triangle you see the big triangle yes we have a smaller triangle that's this one so we have big triangle we have small triangle and then we have sector do you agree with that so what we want to do is consider the areas we have big triangle we have sector and we have small triangle can you tell me how you find the area of a triangle it's area of a triangle or base times height over two yes one half Bas let's look at the big triangle what's the uh what's the base this is one times what's the height instead of Y let's say Tan x and to find the triangle I divide by two I need to show a hand see if you're okay with the area of the large triangle base time height over two we have base 1 height is y yes but y = Tan x I'm going to use Tan x cuz I want everything in the same variable all in x's and then divide by two uh let's do this small triangle then we'll talk about the sector small triangle that's this one right here not this too small one but this one right there are you with me on that one it's that one and that's the big one okay and then we'll talk about the sector in a moment uh what's the base of my smaller triangle what's the height the height of my smaller triangle sin x not cosine X right cosine X would be the sin x would be the height of the smaller triangle because that is actually a point on the unit circle do you believe me so that would be sin x and then well oh shoot sorry I'm off point 1 * sin x over two are you okay with big triangle and small triangle you all right with that one do you know how to find the area of the sector what are we talking which which one of these we talking this is the sector High the area the sector minus the small triangle sure how do you find the area of sector use a big triangle M there's actually a formula for it it's this it is your radius times your angle radius angle two you ready for the math magic ready you have to buy into one thing you have to agree that the area of the sector is in between the areas of the big triangle and the small triangle would you agree with that it goes big triangle then sector than small triangle true so what we're saying here is that the area of the small triangle is less than the air dector Which is less than the area of the big triangle agreed or not what can I do to the twos If I multiply all those inequalities by two which is legal what happens to the twos twos are gone TW are gone so what I'm going to do is change this into s x is less than x is less than t x some basic algebra basic algebra you okay with that so far now here's the cool part what I'm going to do I'm going to divide each of these three things by sin x where sin x is positive it's in the positive quadrant that's okay we're talking about a positive angle right here what that means is that we're not going to change around these inequalities so then this does that you still okay with that so far we divide everything by sin x how much is Tan x over sin x say it again 1/ cosine 1/ cosine because Tan x is s over cosine yes divided by sign that means you flip multiply you're going to be getting rid of those signs so you're going to get one over cosine we're almost done here's do I'm going to reciprocate each of those fractions I'm going to reciprocate the run the run the one the X over sin x and the one over cosine X what that's going to do is also flip around my inequalities if you reciprocate that it's going to flip your inequalities does that make sense to you that's a mathematical truth so I'm going to have one still because the reciprocal of one is one is greater than sin XX which is greater than cosine X over one do you see this thing anywhere you see it right in the middle do you see how it's squeezed between two functions it's squeezed between one and cosine X does that make sense now here's the theorem it's called The Squeeze theorem The Squeeze theorem says if the limit of this goes to something and the limit of this goes to something then the thing in the middle has to go to that same limit if these are equal does that make sense to you it says I'm trying to take the limit as we we go to zero agreed so then the limit of one what's the limit of the number one as X approaches zero can you tell me that what's the limit of one doesn't matter what we're going to right the limits one tell me what's the limit cosine X as X approaches Z Now what do you get of cosine of0 what's cosine of 0 cosine Z is it's not zero one it is one here's what that says check it out this is by The Squeeze theorem it says that right now if I take the limit sin XX as X approaches zero it was squeezed between two functions whose limit was one as we went to zero says the limit of one is one says the limit of cosine as we approach zero is also one what does this limit absolutely have to be if it's between those two functions the limit has to be between those two functions only they're going to the same number what's that limit has to be one it's squeezed it's saying the right bound is one the left bound is one can it be anything different than one no no it has to be one Tech technically I'm supposed to have these throughout the whole thing and that's the big punch line for this part right now we're not done with no so again what we just did we learned we learned one thing about this we learned last time that the limit as X approaches zero of sin x overx was equal to what was it equal to do you remember okay and again for the for the last time we did not prove the squeeze theorem we used the squeeze theorem to prove this thing we bound it between two functions whose limit was one therefore this was in the middle of it that limit had to be one as we approach zero so this equal one that's number one thing you got to remember you got to remember that one that's going to be in your head all right so memorize that thing when X approaches zero of this function you get one if x approaches anything else besides zero well it doesn't matter you can just plug that number in okay that's fine now there's a couple of them that that we might also want to know is there a relationship for cosine firstly and is there a relationship for tangent we're going to look at those today then I'll show you how to put everything together and do any limit well that's reasonable of some trigonometric functions you ready for it yes it's no Monday excuses this time it's not Monday it's Monday Monday no no it's Tuesday I got you Tuesday and Wednesday Friday it's Friday okay I can understand but no no no it's Tuesday today we're rolling so number one thing another number one thing can we find the limit as X approaches zero of this 1 - cosine X over X well let me ask you the question can you just plug in the zero if you do plug in the zero what do you get you get z z what's cosine of Z folks you didn't know that cosine of 0 is one so on the numerator you get 1 - one that would be zero right and of course if you plug in zero X would become zero you get 0 over zero that's a problem can you factor one minus cosine X and cross out the X on the bottom nothing we can do with that we're GNA have to find some way around it just like we did with this one are you ready for it yes it's not going to be nearly as time consuming or mentally consuming as this one was okay this one's actually quite basic here's what I'm going to do what I'm going to do is find some way to to see some sort of an identity in here so so what we're going to try to do is work around this 1 - cosine X and one thing I can do perhaps multiplying by the conjugate would would help us remember what the conjugate is just changes what about that yeah that's it so in our case the conjugate would be one + cosine X over 1 + cosine X okay well let's keep this going then I've got a limit as X approaches zero the denominator is now x * 1 + cosine X you guys okay on the denominator I haven't distributed anything I'm just putting those together for right now now could you tell me on the numerator on the numerator let's do a little work here what am I going to get If I multiply 1 - cosin x * 1 + cosin x what would you what would you get out of that okay try it on your own if you if you can't do it in your head try it on your own distribute it foil it out you know you've got this right you know you've gotve done that uh the first thing you should be getting is a one yes then you're going to get plus Co sin x and you're going to get minus cosine X do you see that what's going to happen to those middle terms they're gone so we we're going to have one and then lastly you're going to do cosine x cosine X is that going to be a plus or minus minus what cosine not just cosine x cosine oh now here's the deal using your massive knowledge of trig identities some of you have and some of you don't have right now to be the Brut honest truth uh what is 1 - cosine 2 x equivalent to using the Pythagorean identity oh you do have that knowledge that's awesome very good if you know that that's right because listen if sin^2 plus cosine s = 1 if that's true which it is true and if I try to make this out of this thing that would be subtracting cosine right from both sides if I do that we make up another identity we make up the identity 1 - c^ 2 X is equivalent to sin^2 X therefore this thing is equivalent to s^ s x can you follow so far now we're going to do a little bit of algebraic manipulation I have gone from this I multiply by the conjugate I've used the Pythagorean identity in a unique way to get sin^2 X I haven't done much with this at all I just said x * 1 + cosine X you follow so far now because this is a limit I can split up limits by multiplication true so I'll show you every step by step watch what we could do here with our sin squ x I know that I could make this limit of sin x * sin x is that not still sin^2 X okay again as X approaches zero and then this thing I'm not going to distribute that I'm going to not distribute it for a very specific reason check this out this is the same thing as X Time 1 + cosine X so I haven't really changed that at all whatsoever you still okay with this you sure raise your hand if you are if you all right with that good okay basically I'll just wrote here's the idea look at because we can separate limits by multiplication check out what we can do I can group together because it just fractions this thing and this thing do you see it does that thing look familiar yeah so if we group it then this is going to be the limit as X approaches zero of sin x x x time the limit as X approaches Z of sin x over 1+ cosine X very brief recap very brief we multiply by the conjugate what that does is creat a sin s x for us the sin s x can be split up into sin x * sin x the x * 1 + cosine X we don't change that at all what we do is we group it in a certain way so that we can use a previously known identity that we have proven that's okay we're using something we've already proven so now we have a limit of x sin xx and a limit of sin x over 1 plus cosine X it's okay to split up limits by multiplication I taught you that property are you still okay with this proof so far all right can you tell me how much is the limit as X approaches Z of sinx beautiful we just did that right we just did that that's fantastic so this is one times now what I want you to do is I want you to try to evaluate this limit at zero are you going to have a problem with that are you going to have a problem why not why don't you have a problem now what's sign of zero good is it okay to have zero on the numerator yeah sure does it to make zero on the denominator no cuz you have 1 plus how much 1 plus 1 that's two you have 0 over two this is 0 over two I agree so how much is 0 over two what's 1 * 0 don't say one how much is one * z z what that says to us is we've just proved something else we now have this identity that's identity the first one we did number one this is the second one the limit as X approaches 0 1 - cosine XX is z interesting you know maybe you can remember it this way well I do at least I remember it this way so something involving s gives you one something involving cosine gives you zero that's kind of interesting right because usually it's if s gives you zero cosine gives you one right if cosine gives you One S gives you zero it's almost the same idea here the S one is giving you one the cosine one is giving you zero it's kind of cool right same sort of relationship is how I remember at least would you like to do the tangent one tangent one's even a little bit quicker actually much quicker let's try to find Tan x overx now could you just plug in the zero what's tangent of zero what's tangent of zero zero because sign is zero and cosine would be one that would be zero but now you have it over Z so you have 0 over Z again so we do need to manipulate it but here's what we can do how much is T as the identities say it is so over cosine so we could make this that are you okay with that one so far if we do a little bit of work with this just a little little bit remember this is like uh X over one you're going to instead of dividing reciprocate and multiply true so this would be the same thing as the limit X approach zero of sin x over cosine X time 1/x still true still true now all in going to do because of the commutativity of multiplication and what we can do with fractions I'm going to Interchange this with that is that okay with you just going to flip them why does it matter the order in which we multiply and this is fractions right so I can make everything one fraction true so that means I could make it one fraction commute the denominators and then split it back up again so that means I okay that's the limit X approaches zero of sin XX * 1 over cosine X I've just switched those two things and that's fine you can do that because multiplication is commutative and you can make one fraction out of that by multiplication see anything interesting have you seen that we're trying to get this thing all the time because we know what that is do you see the sinx overx we can still split up the limits so this would be the limit X approaches 0 of sin XX * the limit X approaches 0 of 1 over cosine X interesting interesting hey now that you have this memorized I hope you do what's the limit as X approaches zero don't read the board right is right up there just kind of hopefully you memorize it already limit as X approaches Z sin XX what is that so this one is one times can you plug in the zero now sure what's cosine of zero it's one yeah what's one over one okay what's 1 * 1 you said one a lot today haven't you that means the limit X approaches Z of tan x x = 1 just like that just like that one did interesting like it so far there's basically three things we need to know these are three limits we need to know of we need to know the limit as X approaches 0 sin XX 1 we know the limit as X approach 0 1 - cos x x that equal 0 and tan XX = 1 again if you know those limits you can break down all the ones I'm going to give you into those three identities if you can do that that this is is very easy to master but it involves some thinking outside the box box in certain cases would you like to see some examples of how to do that okay let's start over here just memorize those three properties of of limits all right those are the proofs that we just did also I'll say it's got to fit those pretty darn perfectly in order to be true all right so let's start with some examples you're going to see some interesting mathematics this is kind of thinking outside the box we're trying to make these things fit in the format that I just gave you okay one of those three formats of of limits we'll start there now firstly is this exactly like the limit that I gave you what's different about this two is a problem two is a problem what we need to do somehow is we need to make the inside angle the same thing as the denominator so for instance if I have a 2X inside my angle I need a 2x on the bottom of my fraction you with me on that you got to have that otherwise it's not exactly the same you can't do that limit I'll show you how to do this in just a second so since we have this maybe we can do something a little bit special the only thing we can really do is multiply by one right that that's it otherwise we change the value of the limit but maybe we can multiply by one in a special way for instance what if I said you know what I want to multiply by one but I'm going to make it 2 over two is that legal to do okay let's see what we can do with this thing now you can put the twos anywhere you want to provided you don't change this a lot of people when they're first starting out in trigonometry hope hopefully this is not you they go well let's just pull the two out front wouldn't that be easy two time no you can't do that that you can't do that right that that is 2x you can't change that unless you use an identity to do that the double angle or the half angle or what whichever ones that you you can manipulate those the only ways you can change your sign of of the inside of the angle okay you can't just pull out that two it doesn't work but we can choose to put these twos anywhere we want so what I'm going to do is I'm going to say all right let's be smart about this now this two maybe I put it in front of my sign but this two let's make that 2x you okay with that so far you see where the twos went now here's something cool do you remember that anytime I have a function multiplied by a function I can split off the limit of that function we we've done that a couple times right here right we did that but with a constant the limit of of a constant is always just that constant let me explain that again when you have a limit of a function times a function you can break it up the limit of a constant is always the limit I'm not teaching anything we haven't had before what that says in plain English is you can always pull out a constant to the front of a limit because the limit of a constant is that constant you are you with me on that so basically pull the two out that's what you can do that two right here I can bring that up front so two times this now some of you might be thinking well wait Mr Leonard why why didn't you also pull this two out why didn't you do that well if I if I had what's 2 over two then I'm right back to where I started right that would be silly I'm doing this on purpose purp so I can make these two things look identical and multiply it by the number that's basically just left over raise your hand if you can follow that feel okay with it all right now here's a cool deal how much is s of 2x over 2x you know how much is sign of limit of uh s of X overx as X approaches zero one one now here's why this is also one you could actually make a substitution check it I'll show this only one time just so you see it once but let's make a dummy substitution like U = 2x you okay with that tell me something as X goes to zero does U also go to zero yes yes it does because if you plug in zero here you get a zero there as well right that means you can make the substitution for the limit so if I make this substitution then this limit now becomes U is going to zero s of well instead of 2x I put U instead of 2x I put U do you see how that now fits our identity perfectly you can do that provided your your b or your your variable still goes to the same spot it's still going to zero because when 2x is zero U is also zero it's still going to the same spot does that make sense to you so we can do this this go oh yeah this is one therefore this is one that's our substitution that's kind of neat so then we have 2 * 1 what's our limit how much two two two can you almost see it though in the original problem s of 2xx how much is it not one it's two what's inside the angle besides the X two interesting isn't it are you ready to make these a little bit more advanced a little bit more advanced start build them up a little bit this was very basic very very basic we're going to start incorporating some other ideas in here so how about just little by little though don't worry now but I got to warn you I'm going to cut out some of the steps I've already covered in the class so for instance I'm not going to ever show you the squeeze theorem for sin x over anymore CU we've already done that I'm probably not going to show you this whole routine for getting that answer anymore does that make sense to you probably not going to show that to you anymore um maybe one maybe once more but after that I'm just going to assume that you you can see that and then you can get there on your own is that fair I hope so cuz that's what I'm going to do oh well this is kind of nasty does this look like any of our identities so far we've only got three of does it look like any of them you got to make it fit one you got to make it fit one of those which one's it's closest to the tangent one the cosine one or the S one it's closest to the sign one but I need to have s over an X true okay is there anything we could do specifically can I multiply by one one something numerator denominator that's exactly the same that's going to give me something over X something over X stunk job remember that you can incorporate new variables as well provided that you multiply or you divide and gives you one I want this thing over X and this thing over X that's what I want can you make it happen here x overx close very close so we know we're going to need to incorporate another X you agree with that right somewhere the x is going to have to happen oh okay so if I listen If I multiply by just an X I get X sign right I don't want that I want overx how we get overx say it again how about 1X is that legal to do is this still one okay let's see what that does to our problem if you multiply s of 5x * 1X you're going get S of 5xx are you okay with that algebraic step what's this one going to be so we're we're multiplying in such a way that we can find some resemblance to an ident identity that we already have here we've chosen to multiply 1X over 1X which is legal to do because this is basically just one it's just one we're doing it so that we get sin x overx you seeing the point here you sure now are you going to be able did were you able to think of that on your own no but now you can use it right now you can use it in some of your problems hey using what we found out here can you tell me what the limit of this thing is going to be why is it five could I do the same thing here um remember I can separate limits by division right so I write limit over limit I can do that so if I multiplied 5 over five like I multiply 2 over two ultimately I'm going to have five times this limit where the 5x and 5x is the same I'm going to have 6 times this limit where the 6X and the 6X are the same I'm going to have that if you want to see it for the last time this is what it would be you'd say oh okay let's do 5 over five let's do 6 over 6 I'm going to do a couple steps at once right now I'm going to break those limits up I'm going to say okay I want the limit 5 sin 5x over 5x I want limit 6 sin 6X over 6X can you make it that far you guys are right with that one yes no what can you do what can you do with this five and with that six what can you do with them outside the limit you can because they're constants and we know that the constants really don't affect the limit because you can separate by multiplication and you'd be okay this is all multiplication that's great so this would be now 5 * the limit s of 5x over 5x as X approaches 0 notice how I'm still writing the limit all over 6 * the limit as X approaches 0 S of 6X over 6X do you see what we've done we basically put two ideas together we put this idea that I need sinx over X we put this idea that I can multiply by some constants and manipulate them in order to get my exact identity out of that now I'm not going to show you the substitution again basically I want you to know that a substitution is possible in these cases how much is the S of 5x over 5x how much one this is this is one right here this limit is one whenever your angle matches your denominator with that sign and your limit is going to zero that angle or that limit is one how much is this limit right here so you have 5 * 1 you have 6 * 1 what's your answer 56 we have that made sense to you you okay with that one all right good you want to try a few more it's kind of fun I like them teach you a little bit about trigonometry won't it or at least refresh your memory about someon and learn some things you can do with limits which are pretty cool oh let's see about this does this look similar to one of our our limits that we already knew how to do is it exactly the same though what's different about it it's got X squar now again can you change the inside of your limit or I'm sorry the inside of your sign can you change the inside of your sign so basically can I change the x s never can't change that thing unless you use identity that's the only time you can do that you can't just say I think I'll pull an X out and put it there no you can't no no you don't what you want to do is make it so that if you haven't noticed this in the previous two examples make it so this angle matches your denominator what do you need to multiply by to make your angle match your denominator cuz that's what we did here multiply to make your angle equal to your denominator multiply to make your angle equal to your denominator that's where the five came from what do you need to multiply by X overx would work so keeping with what we know this x I can't do much with that besides put it out in front of my limit so right here I'll do okay I'm going to put that out in front but these two yes that's what I want because what what I want I want that angle and that denom to be exactly the same you all right with this so far now because limits are separable by a multiplication I can say well this is just multiplication I can separate off a limit of x so this would be a limit of x as X approaches Z times the limit sin x^2 x^2 as X approaches 0 hey tell me something what's the limit of x as X approaches zero it's not x z zero because you plug that in you get zero times what's the limit of sin x^2 X2 as X approaches zero one yeah because you could make that same substitution like we well had on the board earlier for a dummy variable equals X2 right and then as x s goes to zero so does that variable and so does this uh this variable it would be like a you would be going to zero as these two things also went to zero that matches our our identity perfectly this goes to one and as I've said anytime your angle matches up with your denominator you can do that so you pull out that absolutely because it's not a constant it's not a constant yeah the only reason that that's a great point the only reason why we were able to pull this constant out and leave this constant is because if you did this watch this if you did the limit as X approaches zero of that constant what's the limit of two no matter what you're going to right that's why we can do it okay let's continue on with this what's 0 * 1 our limit of x2x sin x^2 x as x 0 is0 we'll try um try a few more of these I really want to make sure you get the handle on this because there's a lot of different combination and problems we can do so I'm going to give you as many as I can in the time that we have that way you feel a little bit more comfortable on your homework you ready for them so again some other examples hey by the way is this the same thing as this this the same as that s of x s is that same thing as sin^2 X what's sin s x mean good maybe you can see where this is going so that's true you're always trying to break it into limits you know limits you understand what's the limit of sin x/x this is going to give you one what's s x what's the limit of sin x as X approaches zero it's zero because when you plug in the zero you get zero so this is ultimately going to be 1 * 0 or 0 this limit if I I'm not showing you a step right here I'm not actually breaking them up if I were to break them up the limit of this would be one limit of this would be zero 1 * 0 is z that one now we we're going to talk about this I'm not going to prove it to you why it doesn't exist you can you can uh get there on your own or or you can look I think the book has a proof in it it's pretty good I'm going to do this graphically just to show you what this looks like now of course sign is an oscillating function right goes up and down up and down up and down one /x as you get close to zero goes to Infinity right it goes there very fast so what happens with this graph it looks like this sure it tops out at one and it pops out at negative one because sign is bounded in those ranges this looks something like this starts off nice it goes down it goes up and down and up and down and up and down and as it get closer it goes faster and faster and faster and faster and faster and faster faster till a bunch of crap happens it's by the End by the time it gets to zero it's like that does it ever get to zero no you tell me can you plug in zero no it never gets there it's undefined at that point you can't do it cannot do it so it just goes nice and slow and then faster faster faster faster faster F until it gets nowhere the other side looks like this as we get closer to zero are you approaching a certain point or is it just continuing to oscillate like that it's not actually reaching a point it's not gonna it's not gonna ultimately go oh I'm gonna I'm gonna level out and make zero or I'm gonna level out and make it's not doing that as you get closer it's still oscillating even faster and faster and faster by the end it's just crazy all right you can't even you can't see it it's going so fast so is the limit going to exist answer is no no it doesn't exist that one does not exist sign of one re does if you ever see that just sign of one of Rex that doesn't exist okay so question this is an interesting question if that doesn't exist Does that exist does that exist oh as we approach let me get my white cat and smoke it did you see that commercial with that creepy guy with the white cat I don't know what that was yeah it was but it was like on the Super Bowl I think it was CEO you know who CEO iseven I Was Made I was forced to watch The Voice last night that's the only forced forced yeah not like handcuffed or anything but like it was on and of course if it's on I have to watch it so it is anyway does that exist let me show you something again we're going to use the squeeze theem are we going to prove the squeeze theem no no we're going to use the squeeze the check this out this is interesting uh do you agree that oh by the way I did make a mistake on the previous squeeze theorem using the squeeze theorem um I was supposed to have the equal signs underneath it I neglected that for some reason I don't know why I did that but with your squeeze theem you do have to have those symbols not those symbols I think I set it towards the very last bit of the proof I said you know what I'm supposed to have these everywhere but I don't know if you caught that let's use some of our knowledge about sign to create some bounds on sign forget the one /x part for a second what's the maximum value you can attain with sign what's the highest sign goes one what's the lowest sign goes so you agree that's true right true okay we'll check this out I'm now going to create this in here by multiplying all three of these sections by X is that legal to do sure yeah so if I multiply by X actually I think I yeah I do I want the absolute value of x because I don't want to change those signs so I'm going to make sure I multiply by the absolute value of x okay so here's a slightly better interpretation of what's actually going on in the problem we're working on right here we're trying to work on the limit as X approaches zero of sin 1X * X now of course we know that sign is bound between 1 and - 1 s doesn't get any bigger than one doesn't get any less than 1 so we can go naturally make this an equality any sign function is between -1 and 1 including s of 1 /x so we have that down can't get bigger than one can't get less than one now the way we make the jump from here to here which in class I have absolute value around this X and I said we can get there by multiplying this absolute value of x and this absolute value of x and this absolute value of x which is true but it doesn't quite give us back exactly what we want which is X without the absolute value how we get there is just by a little bit of little rational thought so we say if we multiply by X in all three spots what could potentially happen is if I don't have these absolute values I could get a negative number over here which wouldn't be an upper bound so what we're saying is let's make this one absolute value of x that way we know that that's always going to be positive no matter what I plug in I have positive X well that's certainly going to be bigger than x sin of 1 /x if this is true then this has to be true where X is positive also if we let X be negative we're going to have to have a lower bound now if I don't have this absolute value what that says a negative times a negative could actually get me a positive I don't want that to happen I want a legitimate lower bound so that's where the absolute value is coming from that says all right well then x s of 1x certainly has to be bigger than that negative number if this is true then this one also has to be true and that makes our double inequality here from there it's a simple matter of limits we say well we know the limit as X approaches zero of the absolute value of x that's going to zero we know that that's that's that V curve as we're going from both sides the height that function is getting down to zero so that limit exists it's zero same thing the limit of negative absolute value of x that's negative absolute value of x we're going up to a height of Zero from both sides that limit zero so we we have the limit of this function is zero the limit of this function is zero since we've now squeezed our function we wanted to find out this one since we squeeze that between two functions whose limit is zero as we approach Zero by The Squeeze theorem we can now draw that conclusion that's how we get the limit as X approaches Z of x s of 1 /x equals z does the limit only exist because you have an X in front of the S one /x yeah okay and which is interesting because that little that X right there in front of it allows you to bound it between two numbers and then use that fact to say the limit of this goes to something Li of this goes to something the problem is if you look at this one just thetive one sin 1/x less than one where's this limit go where's this limit go are they the same then you haven't squeezed it that's the problem we need that X to say okay by multiplying by well the absolute value of x which which is a small point but multiply by absolute value of x the limit of the absolute value of x yes that is going to zero and the absolute value of x yes that is going to zero that is what actually squeezes it that's the point without the X you can't do it it fails here it doesn't work here if works here this is another one that we can use limit of x sin x or sin 1X that does exist it actually equals zero it's an interesting thing just by having that X in there so our whole idea for this is you're trying to break up these limits into the identities that you the limit identities that we can work with the ones we've already know that's the sinx overx goes to one and the 1 - cosine X overx goes to zero and the Tan x overx goes to one again that's what we're trying to do um I think I have three or four more examples would you like to see some other things that we can do okay I'm gonna I'm going to have to kind of paraphrase these for you because I just want to give you the the head start on them some of these are are in your homework as a matter of fact of think but I want to give you some key points so that you can do these on your own so I'll be moving kind of quickly uh follow along if you're not quite getting to see see me in the math lab I'll be there often or my office hours or check out these videos again later on see that you can follow these on your own you with me on that so A couple of the things that we can do this is basically just practice at this point you've learned everything we need to know okay while this looks kind of nasty I want you to see that we can actually do a lot of fun things with this firstly do you see how we have a few cosiness up there we're probably going to be moving towards the cosine limit that we we learned about so if we do that well what I want out of that limit is the oneus the cosine true overx I've already got the overx but I don't have the one however what I can do with that number two can I separate that into 1 + 1 is 1+ 1 the same thing as two sure so then this is really check out what I'm going to do I'm say this is really 1 + 1 I'm going to reorganize the stuff then and say this is the limit as X approaches Z of 1 - cosine 3x + 1 - cosine 4X all over X does that work for you is that still the same exact thing just reorganized it now what's cool about fractions is that if you have a common denominator you can add them right and subtract them what it also means you can pull them apart by that common denominator so what that says is I can actually make this 1 - cosine of 3x/ x well there's one of them but now I have the common denominator plus 1 - cosine of 4X x wouldn't those be different limits though and then you're adding them together we are adding two different limits right now we're not there yet but that's a good point we can split up those limits by addition right so we now have two different limits oh now what can we do now what can we do we can do that what we want is the angle to be the same same as our denominator so if we do that if we do the 3 over3 thing and the 4 over4 thing move those numbers out front don't affect this one minus cosine of that that uh that angle then we'll be okay then that's going to allow us to do that substitution just like we did for the sign did you follow that so we want 1 minus cosine of an angle over the same angle basically down on the denom that's what we're looking for here so if I multiply this by * 3 over 3 * 4 over 4 all this becomes is 3 * the limit X goes to Z of 1 - cosine of 3x over 3x plus 4 * the limit 1 - cosine of 4X that's hard to draw 4X so we do the same exact idea that we did for our sign only here can you tell me what's the limit as X approaches Z of 1us cosine of some angle over that same angle how much is that that's the zero so we have 3 * 0 plus 4 * what's that one going to be we do that substitute again just like we we learned for signs so 3 * 0 4 * 0 what's the answer I know right all that time spent and nothing zero you should should have seen the looks on my my math C kids when I I did a whole problem using rational functions took about 10 minutes and then the solution didn't work out because it made the denominator equal zero and they're like wait there's no solution like yeah isn't that exciting they're like I hate you oh well okay other things you can do this is very similar uh Now understand that we were able to break up these fractions because we had that denominator we can do the same thing here so we could actually make this limit as X approaches Z of x^2 /x - 3 sin x overx you guys okay with that one yeah we can do things like that we can split up our limit using our algebra how much is x^2 x so we'd have a limit as X approaches Z of x - 3 * sin x/ X tell me what happens here tell me what happens where does The X go to Zero by the way the reason why we can do that is because we're not actually getting to zero right we can actually cross out the X and X that's okay so what's the limit of x is X approaches zero this is zero minus what's uh what's the limit of three is the three going to change now you you know I'm skipping steps here right you know that technically what we're supposed to do is is separate these by subtraction then separate these by multiplication but we found out that earlier with our pols if you can plug it in and you know the identities then you're going to be okay so here this goes to zero this three that's going to stay at three what's the limit of sin XX as X approaches zero what's my answer answer is3 would you ra have be okay with this one we'll practice a couple more next time I want to show you two other things we can do and that'll wrap it up so we're still learning how to do some problems involving trigonometric limits so when we deal with these limits our basic idea is you want to get it down to one of those three identities that we knew for limits uh namely sin x/x as X apprach to 0 goes to oh boy one one very good okay somebody else one minus cosine x/ X as X goes to Zer goes to zero and tan XX oh good all right you got that one that's that's the one we don't use that often either oh nuts okay so basically we're trying to get those things in these problems some way or another now often times you'll have to use some sort of an identity to do some of these things so for instance when I look at this this top example t^2 over 1 - cosine 2 T if I plug in the T I have a problem because I have 0 over 0 do you see the0 over Z that's a problem but is there a way I can manipulate this in such a way to get rid of the 1us cosine s t that right there at this point for you who have taken math 2 our trigonometry you should that should be going oh yeah I know what that is what's 1us cosine squ T do you know sin Square t very good so I could change that now at this point there's a couple ways to go about doing this problem one option is do you see how I could break this up T * t s * s do you see that you could have t over sin * T over s you with me you can break each of those up as different limits each one of those limits would be going to one I I didn't say this to you but it should maybe have been apparent to you from the squeeze theorem that this limit is also equal to one was that apparent to you as well that's the same thing that's the same thing the reason is because when you do the squeeze theem instead of reciprocating if you left it alone you have that already that's that's in there when we proved it with the squeeze theorem also I can show it to you this way um do you remember that if you take out an exponent or you reciprocate a fraction you can actually have a negative exponent some of you zoning out already let me let me show you what I mean maybe let's make more sense if I flip these around but make it negative 1 that's the same thing as that do you remember that reciprocation of a fraction changes the sign of your exponent if that's to the first Power I can reciprocate those fractions make to negative first Power yes I can pull out a negative I pull out any exponent right so I could have a limit of sinx overx that's one 1 to the 1 still gives you back one either way that's one so in our case up here we could do it that way that I explained earlier we could break this up as T sin T * T sin T each of those is going to give us a limit of one are you with me on that one * 1 our our ending limit this should equal one but I want to show you the different way to do this that's one option do you see the option that you do that the other option is well if you did something similar with that exponent I could say that this is the limit as T approaches zero since the numerator squared and the denominator squared I can break that exponent out to the top you with me on that one sure okay also if I change the sign of that exponent I can reciprocate my fraction do you follow that one so this says all right well then why don't you just make it limit T approaches zero of sin t/ T that's something we want but now since I reciprocated the fraction I have it to the negative2 power what we can do with the ne2 power is pull that exponent outside of our limit now we're home free do you need me to recap here a little bit Yeah okay recap still get a kick out of that every time trig identity youall need to know that one taking an exponent out it's very similar to this if you have x^2 over y^2 that is X over y squ agreed then this is similar to this if I reciprocate the fractions y /x then that's just -2 that's exactly what I have done here and then there true properties of limits say that if you have a function raised to some power it's the same thing as a limit of that function raised to that power so basically says I can raise the limit to the power instead of raising the function to the power and then taking the limit so in other words that's possible pull out the exponent take the limit first first and then do the exponent you okay with that one that's the recap now what's the limit as T approaches zero of sin t/ T so you all agree that's one right what's one to the2 power one it is just one yeah anything to one to any power gives you one so this is going to give you one you okay with that one well that was fine any questions on the first one is it only one if it's approaching zero or can it be approaching another number that's a good question if it's approaching any other number besides zero well then okay the only difference on this one for like right right here if it's approaching number besides zero well then s is actually a number right so you can plug that in and be just fine you wouldn't have to do any of this uh the only time that really matters is if the S of T is zero which happens at Z and then pi and then 2 pi again for this example here it would really matter unless T goes to zero so that that's our our situation so if T any if T is anything else besides zero you don't even have an issue remember I said you should always plug in the number first to make sure you even have an issue so if this was like going to one or well let's make it better Pi / 2 you go let's let's do pi over 2 do I need to have a limit Identity or anything no you just do s of pi over two over pi over two and that will actually work out for you be 1 over pi over 2 or two over Pi that would be your answer so you wouldn't have to do any of this stuff if you have a number besides zero are you with me on that it all goes back to you should try your limit before you start doing fan this is fancy stuff try your limit before you do fancy stuff plug in the number if you don't have a problem then that's your limit if you do have a problem then do fancy stuff fancy stuff my fancy hands okay does that answer your question though yeah all right let's try the next one now let's practice this then are you going to have a problem if you plug in zero What's the numerator become zero oh good we got that one what's the denominator become this is pi over two right pi over 2us 0 is pi over 2 right what's cosine of pi/ 2 I'll give you a hint start with the Z and rhymes with Hero Zero yeah it's zero you get zero over zero that's a problem so to answer your question again would you could you if you if that wasn't zero if it was something else could you potentially just plug in the number without doing any of this fancy stuff where about to do absolutely but now that we have 0 over Z this is where you have to use those limits and identities to get something you know what's the things you know you know the limit of sin x/x as X approaches zero is 1 you know the limit of 1 - cosine XX as X approaches Z is z and you know that uh the limit of tan XX as X approaches 0 is one again that's the only things you know so if you can't do the limit directly you try to make it into one of those things you follow me on that okay so what can we do someone who's really good at identities tell me what that is I'll take care of the numerator for you all right here you go boo do the denominator what's cosine of 12 Pi minus X it's a special little thing half angle identity yeah oh man you guys are killing me you guys killing me know your double angles your half angles your pythagoreans and the tangent ones okay know what secant cosecant cotangent mean and you'll be okay those are the basic ones you need to have those down what this says if you want to check in the back of your book cosine of 12 pi x or minus your angle whatever that that happens to be whatever variable you have that's equal to sinx that's it this is sinx did that make your problem easier much easier right if you try messing around with this one you're like I have no idea what what's Mr Leonard going to do there this is crazy stuff but that's actually just an identity do you know how much this is limit of x over sinx as X approaches zero it is one yes we actually just talked about that one one and done cool yeah okay so do you sometimes need some identities to do these problems yes you do if you be good at identities in this class not a professional otherwise you'd be teaching but relatively okay okay you need to at least know them and how to how to use them that's why I had you do that that identity section right so that was heads up on you should do this now most of you skipped those problems don't problems uh mostly you skip those problems but you you should have the idea that yeah we're going to be using this in class and it's somewhat helpful unless you just want to spend massive amounts of time banging your head literally against the wall and pulling your hair out I don't want you to go bald so learn your learn your identities and then you get a little bit easier okay now our last one that we're going to do in dear again I can't give you every example but I've given you quite a few I think you'll agree what can we do with that what do you think any ideas we did something with 1us cosine earlier in the class maybe that would work conjugate conjugate conjugate yeah let's conjugate it why don't you multiply by the conjugate right now I'm going to walk you through this one multiply by the conjugate of 1 minus cosine Theta do them both the numerator denominator uh again what's the conjugate uh of this that I'm talking what how do you find the conjugate basically change change the sign okay so what is the conjugate specifically for us you got to do it on both the numerator and denominator you okay so far we don't necessarily have to distribute the numerator I'm going to leave that alone in case we absolutely have to later what I do need to do is distribute the denominator you follow so let's try that so we're going to have the limit th approaches Z Theta squ 1 + cosine thet can you tell me how much this is going to give you when you do your your distribution what are you going to get one hey we've seen this already how much is 1 - cosine squ still okay so far are you okay so far with that give me a little head knot if you are yeah okay good do you see anything we can do now what do you think just see anything looks familiar in this problem what looks familiar maybe that we've done earlier in this class that you Coulda squ you see theeta Square sin squ right we just did that it's right here we've done that problem so what I want you to do right now is break that up I want this the Theta squ over sin squ times the remaining well we'll talk about that in a second right now all I care about is that you see that this right here is going to go together do you see it's going to go together and it's going to be multiplied by 1 + cosine th do you all have that in your paper right now you seen how to if you didn't get that you seen how to get that yes no it's okay if it's a no but I need to know it's a no it's it okay or not yeah not so much this is like uh Co this is like over one right if you multiply numerator time numerator and denominator time denominator it will give you back that thing therefore we can split it up uh in the opposite manner okay now what can you do with limits that are being multiplied together can you split them up by multiplication can you split them up by do that now make two limits out of it limit of this one time the limit of this one do that for me also one little piece of information when you're doing these limits you have a lot going on make sure you're putting the parentheses around the argument of your limit just to make sure that you know I'm taking the limit of this whole thing not just a limit of one and then adding cosine Theta does that make sense to you make sure you see that did you make it this far okay now since we've already proven this we already know how much this limit is how much is this limit we can redo it if you really want to right pull out the two reciprocate it -2 uh 1 to the second power is still one so this thing is one now can you evaluate this limit at theta equals 0 is that okay to do now sure you have no denominator that's fine plug in the zero and do that on your own notice how I'm not writing limit anymore this limit is equal to 1 times this limit is equal to 1 + cosine of 0 since Theta is zero notice how you still need par because you're multiplying one * something that's okay how much is cosine of Z it is how much one cosine of 0 is 1 so what we have in here is 1 + 1 so 1 * 1 + 1 what's our limit equal to limit is two there are a couple different ways you could have done this you could have broken this up independently done a few different things this is probably in my opinion the most concise at least one I found it's most concise so do that and then we can do any limit as long as we're finding uh at least some identities that we've dealt with before inside of our limits and using those things how many people understood what we talked about so far our section how about this story are you guys all right with this any questions that we have before we continue all right finally done with Section 1.2 feel good it's good for me this thing was long my goodness