Now we will talk about to limit laws. What's the idea before we get involved in this? Right now the ways that I know how to compute limits are looking at graphs or pulling out the calculator and computing messy tables. Both of them are not as exact as we want. So if you had a graph how do you know the limits one versus say 0.999. You're never going be able to look at a graph and say that no matter how much you zoom in and if you pull out the calculator, same thing, you get decimals that or 0.999 and you can guess that it's going to one or maybe it's going to 0.9998 or something like that. You just don't know exactly and we'd like a better way to do these things and that's where the limit laws come in. So let's set this up. First we're going to pick c to be a real number, any number you want and we're going to let the limit as x goes to c of some function f of x and the limit as x goes to c of some other function, g of x exist. We want to functions whose limit at some points exists. Now here's the laws. This is going to be our rule book for the new game, the new calculus game we're going to play and this is where if you don't follow these rules, you're going to get marked wrong. No one wants that. Here we go. So first off limits, and this is a beautiful thing, it's almost too easy to miss this. Limits distribute over addition what does that mean? If you have a sum, then limits go to the first piece, or it doesn't matter, sum or difference and the second piece. Before I move on to the next one, we need to talk about this. This is too exciting not to talk about. Most functions don't distribute over addition. That's usually where if you're going to make mistakes in algebra or something like that, they're going to say something terrible when we write it over here on the side so I can immediately cross it out. If I have sine of x plus y, you can't just bring this sine in. No, it's awful. Don't even look at it. You have a whole identity from trigonometry that tells you how to do the sum of sine of x plus y, same for cosine. If you have square root of x plus y, you don't get to break this up. Now, oh gosh, I don't even think about it. Breaking, distributing over addition is not something that naturally happens with functions, but it is a beautiful thing. It's when you talk about the behavior of addition or subtraction of functions, limits do distribute. Right away, limits behave our first rule here, nicer than most functions that allows us to do more with them. For all the same reasons if I have a constant times a function like where the function wants to go times a constant, you can bring the constant out front. This is the constant comes out front and you can evaluate the function or take the limit of the function first and then multiplied by the constant. Again, that's not true either for functions like if I have f of 2_x, I can't just pull the two out. It doesn't work this is not multiplication doesn't work like that, for most functions. Limits behave nicely. If you happen to find a function that just has the property that it distributes across addition, maybe got lucky, but limits will always do that. Limits also have the amazing property that they distribute across multiplication. If I have a product of two functions, the limit of a product is the product of the limits. In a weird way, it's hard to do something wrong with limits. Everything is legal for the most part and after multiplication, of course you have division. So f of x over g of x. Now, you can break this up, but this is the only one that has a little catch. You can break this up into the limit of the top one over the limit of the bottom one. There is a catch here because I have a fraction, even the limits with all their power and all their glory, they can't divide by zero. I'll say this is not equal to zero. Remember up here, I just said that it had to exist. If it's zero the last one doesn't work, you still can't divide by zero at the end of the day when working with limits. These are the rules and they have their names, the sum rule, the constant rule, the product rule and the quotient rules for limits and they get us a lot because powers are just repeated addition. Let me just write it first before I start talking about it. If I write something a function raised to a power, so squared, cubed whatever, since that's just repeated addition, then the power actually, you can apply rule number one over and over and over again, distribute this thing. This becomes the limit as x goes to a of the function raised to a power. You can bring limits inside powers. I've got good squeeze it in over here, since you can write square roots as to the 1.5, cube roots to the one-third, roots also behave the same way. If I have the nth root of a function and I want to take it's limit, I can bring the limit inside. What I did here. I can take the limit inside first, and then I can work with underneath the radical. Hopefully keep these handy. This is the rule book for limits. This is what we're going to be allowed to do and given these rules, I'm going to go off and find lots of limits. We'll use these in the next example. Let's let the limit as x goes to two of some function before, and let's let the limit as x goes to two of g of x be minus two, and we'll let the limit as x goes to two of h of x be zero. Here's I want to find: I'll put it down and then we'll go through and talk about them to f of x plus five g of x limit as x goes to two of g of x squared limit as x goes to two of the square root of f of x. Our goal eventually is to find every limit we possibly can. They will get harder and harder. But this is the first application of the rule book here. The thing to appreciate is that you're not allowed to do most of these things with regular functions. If you do, your algebra and precalc teacher will yell at you. H of x, x, that's an x, h of x. Here's a bunch. What I'd recommend here is have the rule book handy. Go through and pause the video and try to work these out using our new limit rules. Then come back and check your answer. Let's solve these in the meantime. You can pause the video if you haven't worked these out. Here we go. One of the rules says that limits distribute over addition, and that scalars come out front. So you can think of this as this guy goes to four, g goes to negative two so this becomes 4 plus 5 times negative 2, 4 plus negative 10, negative 6. Good. Second one, it's a power. Limits sneak right inside of powers. This is where g goes to two. This goes to negative two. This becomes negative two squared, which is four. Power square roots, you can think of this as to the one half, has to limit as x goes to two of the function f of x to the one half, so roots are just powers. That means the limits sneaks right inside as well. Think of this as the square root of four. Well, that's what the limit is as x goes to two. This, of course, is just two. Over here, I'm allowed to bring a limit in as long as the denominator is not zero. My denominator is not zero. This becomes 3 times 4 divided by negative 2, and that is 12 divided by negative 2, that's negative 6. Over here, now I have a problem because if I plug in, I get minus two over zero and remember you can't do that with the limit laws. It doesn't work that way. This one here will say undefined. We don't know. This one here you don't get an answer. I can't use the limit laws for this one. If you [inaudible] come back and give me an answer. This one over here with the g, with the f, I have g go into negative two. That's fine. This thing's going to zero, h is going to zero, and f is going to four. But all together you get zero over four. That's perfectly fine. If you have no slices of pizza, of the four slices of pizza, you have zero slices of pizza. These are some examples that you can do with the limit laws. Again, watch out for question [inaudible] where you divide by zero. One thing we talked about, but let's give it an official name is called the direct substitution property. If f of x is a polynomial or rational function, rational function; friendly reminder, when a polynomial divided by a polynomial and a is in domain of f of x, then to evaluate the limit, x goes to a of f of x is to just plug in. You heard me say this before, plug-in but when you can. This is when you have nice functions, polynomials like x squared, x cubed, these sort of things. Plug in when you can. If you can, then you evaluate the limit. If I ask you just to see an example of this, what's the limit as x goes to zero of x squared plus 5 x plus 7. That's a polynomial. Can I plug in so zero squared plus 5 times zero, is just 7. When you can plug in, you're good to go. When you can't like if there was a division by x or something down here, there's more work to do. But this is our first easy approach to direct substitution properties its formal name, but I'd like to just say plugin when you can. If you can plug in and evaluate this thing, you're all set to go. Let's talk about another function that comes up a lot when talking about into limits and it's called the greatest integer function. I don't know if you've seen this one before, but this is a very specific function. It comes up once in awhile. It's usually written with double bars. I've also seen this written as this way, although our book doesn't write it that way. But it's known as the floor function as well. Which is why they write just the lower pieces as if it had like a little floor. The floor function are also the round down function. This has to do with rounding. Most of you who maybe are surprised to hear this or though I don't know, maybe not. When you are little kid and you talk around usually as say 5 or more, you round up, less than 5, you round down. There's a little thing you can learn like five or more and let it soar for less, let it rest or something. That was what I learned as a kid. Just as some examples with his function, the idea of this function is you always round down. What does that mean? If I have the number, let's use the double brackets. We'll switch it up like 3.1. You round this down always, this is 3. If I had like 3.5, here's where it differs normally I'd round this to 4. But since we're always rounding down, this is also 3. It's annoying to write these double brackets. I usually like to write the floor, but our book does these double brackets around the function. Just to show you how serious this thing is about rounding. If I have 3.999, like that, a super close to 4. But we're not at 4. We round down, always round down. This is 3. It's one of the things where either you there, you're not, I don't care how close you are. If you're not at four you round down, you can throw in some crazy functions like Pi. Pi is 3.14, so that's just 3. Of course, if you plug in 3, that's perfectly fine too. You're already there, you get 3. It also pick your favorite thing. If you graph this function, this is a perfectly fine function to round down. There's also the equivalent roundup function, but greatest integer function, and it's just round down. If you graph this thing, the graph has a cool shape. So think about what this looks like. Let's pick a color. At 0, at all the integers, you're yourself. If I plug in 0, I get 0. If I plug in 1 or 2. So I get 0. If I'm a half, I'm 0, if I'm 1, I'm 0. If I plug in all these numbers between 0 and 1, they all round down to 0. But at one you have an open circle. Because when you plug in 1, you get a closed circle. Then if you're 1.1, 1.2, 1.3, they're all rounding down until you hit to like 2. Then it jumps up again. This is a, they call this a step wise function. So close circle. It goes like this and the pattern continues off to the negative numbers as well. You have open circles on the right and closed circles on the left. You get this staircase, so it's calling a staircase function. These are called jump discontinuities. This functions used in modeling certain things like the post office. If you have, if it's 10 bucks, if it costs x number of pounds, as long as you're under that amount, the price doesn't jump. But as soon as you hit a certain weight, then the price jumps. So used a lot for shipping. These function are apparently five function just may be new to you if you haven't seen this before, put this down as another graph that comes up. It's good when trying to come up with counter-examples. With any graph you can ask about the domain. What's the domain of this function? We only get rounded anything, all reals. Here's a cool one. What's the range of this function? Now you don't get any numbers except 1, 2, 3. So now we get to use our z are integers. That's cool. That doesn't come up that much. It's a great function for asking about limits. This is where we care about more so, like what's the limit as x approaches 2 from the right? You're a little bug and you're walking towards 2 from the right that wants to go, we'll make it fancy do the double bars. That wants to go to 2. But the limit as x goes to 2 from the left. Now that's going to be different. You can see it from the graph. If I come in from the left, now I want to go to 1. Of course they are different, and so the overall limit as x goes to 2 of this particular function, because the two one-sided limits are different, we would say this does not exist. This is a great graph for just giving examples of one-sided limits. So know this graph added to your collection, your repertoire graphs, put it on there. It'll come up more and more as we go through it. Good job, see us, next time we'll do some more examples.
