 So in this lecture I want to take you through seven limit laws. Now if the limit actually exists There are seven laws that you should know about. The seven sounds like a lot But really isn't that quite simple because in an add, subtract, multiply, and divide a few limits But these are proper laws. If the limit exists you can apply them and that will make the calculation of the limits very easy So in this chapter I want to have a look at the limit laws. Here There are seven of them. I suppose they're not really laws, but they do work for us in that regard. One thing about them is Well, while you've got to learn how to apply them, please please please remember they only apply when the limit exists Now what comes first? Do I show you when limits exist or not? Or do I show you the laws first? Well, that's kind of a chicken and a neck thing. So let's go let's go for it this way But remember this only applies when the limit exists If the limit does not exist then these laws don't apply and we'll go through examples and you'll clearly see what's going on Now these aren't difficult at all Let's start with this first one. It says the limit as T approaches some constant C and then you have the addition or subtraction there we have addition or subtraction of two polynomials and That can be rewritten just as the limit as T approaches C of each one of them there We have the f of t and then we have the g of t so you can just write the two limits separately It's as if you're distributing the limit inside of the the parentheses there, but I don't see it that way the limit as T approaches C of the additional subtraction of two polynomials equals the limit of each of those subtracted or From each other or added to each other Second one is just as simple if you have the limit as T approaches some constant of Another constant times a polynomial or times some function. I shouldn't just call these polynomials actually any kind of function you can simply write that as This constant times the limit of the function itself as T approaches some constant so that You can almost say we can just bring that out we can bring out that constant now we have the limit of the product of Two functions as T approaches some constant that equals yes You guessed it you can take the limit of each of those functions and then multiply the result with each other Just the same for quotient Just remember Now that sign this means for all slightly incorrect use of it the mirror for all so just see that sign is for also the limit as T approaches this constant of the f of t divided by the same for the g of t for all cases Where the limit of g of t does not equal zero really? We all know we can't divide by zero Now this one will help us a lot. I have this limit as t approaches C of some function squared So this function here is squared And that is the same as taking the limit of the function and that result that you get you then square it And it's not only just for square at it works for for the powers as well Now the limit as t approaches some constant of another constant equals just that constant nothing changes and then the last one the limit as t approaches C of Just t on its own equals just that constant where t here is just that's just Some function of t isn't it? I suppose you can just write the function of t equals t That's what we have in this instance, and you're just going to replace The t with this constant so they the seven as you can see they are really not that difficult But they help you tremendously in solving limit problems Now the best way to understand these Limit laws is to use them in action Now look at this. We simply have a function here f of t and that f of t equals t cubed minus 2t squared plus t and Remember what we said if the limit exists now you don't yet know when a limit exists or doesn't but let's say that all The examples here the limit actually exists We can just we can just take the limit of each of these so this we could have seen and Let's say this is the g of t and the h of t and Let's just call it the m of t So the g of t was t cubed the h of t was negative 2t squared and The m of t was just t so we can take the limit of each of these and then look indeed This is what we've done We keep we keep saying that the limit as t approaches to approaches to approaches to just as it was there And now we just take the limit of each of these separate polynomials so The easiest way to do this. What is the limit of t cubed as t approaches to well We just put two in the place of t. That's going to give us eight Here we have our negative two and look at it We're actually you making use of one of the other limit laws. We're bringing the negative two Outside and there it is Store that negative sign there the negative sign there and then in place of t We just put two again and remember the limit as t approaches to of t Well, we just replace t with two so it's actually more than just the addition of and subtraction here of polynomials And that's just a simple arithmetic and we see that the answer is indeed 14 Let's look at one of the other laws here We have the limit as t approaches to of three t cubed and here We have this constant k was which is three and we just bring it outside Yeah, as you can see of the limit and then take the limit of this polynomial Which is t cubed now later on you'll see again This is that law where we can just take this power and take that outside So just take the limit as t approaches to just of T which will just be two and then cubit which is eight, but the same happens here The only law I want to show you here is just to bring out the constant outside of the limit and Then still take the limit of this variable. So you can bring that variable out, but you certainly can bring constant Coefficient out and we've done that with a three and Then as t approaches to we can just replace two in there two cubits eight eight times three is indeed 24 Now let's have a look at this in this example I've got two polynomials t cubed and t minus one squared There's one as the other and I'm multiplying them with each other So what can we do? Well, we can just take the limit of each of them individually and then multiply that result so the limit as t approaches to of T cubed well, this is two cubed which is eight and The limit is t approaches to of t minus one or two minus ones one squared is just One and eight times one is eight So all I've done is I've taken the limit of each of these two polynomials separately and then multiply the result so in this example we have the Quotient of two polynomials. There's our f of t in the numerator and there we have our g of t in Denominator there now first of all this let's have a look we're taking the limit then of each of them individually We've just got to make sure that this does not equal zero and in least at least And if I plug two in there, it's two minus one is one squared is one So certainly there's not zero in the denominator and we get to a solution of eight So look at this. We have one polynomial divided by the other. We're taking the limit of that We can just take the limit individually then of the polynomial in the numerator and the polynomial in the denominator And here we've got this example. We've seen it before I can simply just plug the two in and of course That will be eight, but I just wanted to show you when you have a power up there. So this is this polynomial So we go it's this polynomial to the power three t to the power three I can simply take the limit then just of that polynomial and then at the end Cube the result there and I get to eight so no matter which way around I do it I can take that power outside of the limit shouldn't shouldn't use those words, but that's what we do and And then get the limit of just of the polynomial t or you just plug it in and just do it the regular old way works in either case Okay, these things are getting embarrassedly embarrassing the easy We've got the limit as t approaches some constant, which is two of another constant three Well, this has got nothing to do with this t approaching to there is a constant there. It is just three Simple Here we have the our last example very simply the limit as t approaches to just of t just of the variable and We can just replace it with two In there and the answer being to so these last two really are simple. So there you have it all the limit laws So please use them. They're so easy to use but only when the limit exists and Eventually we'll get to when a limit actually exists and it doesn't for now. Just enjoy these limit laws