All right, everyone. Welcome to Math 165, Business Calculus. Um, we're going to dive right in. But before we do that, just a um one thing um that I've learned from uh after teaching this course for years is that it really it's often not the new calculus ideas that trip students up. Um while obviously calculus isn't the easiest thing on the planet, um most of the time it's the algebra that is what gives the students the the most problems. So that's why I've provided you with all those that vast amount of worksheets um for algebra reviewing. I've given you quite a handful of those. Um I'll also be posting my math 130 or pre um college algebra lecture videos as well. Um so that all of that material will hopefully help you revisit anything that needs refreshing. Um because you can do this um but you do want to take that review seriously as we go on throughout the course. Okay. So, my role here is to guide you through this new material, the calculus. And you know, that said, uh I know sometimes we're going to need to revisit some prior material, and I will be happy to do that where it's needed. Um but just remember, you know, I can't review every little detail. So, um consistent review of your own personal algebra skills definitely will go a long way towards your success in the course. So, without any further ado, let's dive right into section 2.4, which is limits. So, just real briefly, what is calculus? Um, I'm not a big history buff, but an interesting fact is that it was developed by two separate people, Newton and Linets. Um, completely independent of one another on different sides. I think one was German and one was English. I don't remember, but two different people didn't know each other. And they actually invented um the same exact concepts on their own. And those two main ideas in calculus basically splits this class in half. The first part of the class is going to be what we call the derivative or differential calculus. And that is going to be finding what we call a tangent line to a curve at a given point on the curve. So for example, if we're given say that point right there, the tangent line is going to be a line that goes right through that point, skims that graph right at that one spot. And in particular, it actually hits that graph right at a 90° perpendicular angle. So once we've drawn that tangent line in there, the goal is going to be what is the slope of that line? As we know from algebra, slope is rise over run. So we'll talk about that. And that is called differential calculus. Now, what does that slope of that line mean? We'll get into that. The second part of the class is called integral calculus. And that is where we're going to be given some sort of curve. And our goal is going to be to find the area underneath that curve. So we want to find the area of some region bounded by an arbitrary curve. So you're going to have some crazy curvy curve and we want to find the area underneath it. And that's called integral calculus. And so that's going to basically break up our semester. Now going back to Newton and Linets, um it's important to know that only because I do find it's kind of interesting that two separate people invented the subject of calculus. They do have different notation. So we'll have these like I said these two big concepts and there's going to be what we call Newton notation and what we call linenets notation. Okay. So the first thing that we're talking about which is differential calculus and more specifically finding derivatives. Before we can find a derivative the first thing we need to do is find what we call the limit. And I'm actually going to start on the next page first. So, I made this a long time ago. You can tell because it looks like it's been scanned 25 times. But this is introduction to limits Megan style. And what we have here is some fancy notation that says a limit is the following. If you have a function, so let's just say we're talking about this graph. If you have a function and you let your x values approach and that's a very important key word approach a certain number c then the y values approach or get very close to a certain number which is called the limit of the function and our notation is limb as our x values approach some number c and if I go over here of whatever that function is equals some sort of number, some output, some y-value. So, a lot of what calculus is is a lot of crazy notation, really meaning things that aren't too overly complicated, but the notation makes it seem complicated. So, we can find the limit a few different ways. The easiest way is just by looking at a graph. So, let's do the easy way first. If I look at this graph and I want to find four different limits, the first one I find is the limit as x is approaching one. It's getting close to one. So obviously there is one and I imagine that I am standing just on either side of one. So I'm standing just on the left, just on the right. And if I'm the person standing just on the left and I look up or I'm the person just on the right and I look up, I'm going to be like, okay, I'm getting close to one. I'm not standing on one. And when I do this in class, I actually stand on a chair to represent I am one, but I'm not quite standing on that chair. I'm just on either end. The limit is well, what does the y value look like it is approaching? So either person here on the left or right looks up and goes, well, I don't know. And it's kind of like this gray zone. I don't know what's happening right at one. I don't know what the function value is at one, but both person on either side looks up and goes, you know what? It looks like the y values are getting close to two. They may or may not be exactly two, but a limit is what you are approaching. So the limit as x approaches one is two. Because on either side of that limit, when those people look up, all they see is what they can see. They can't see what's happening at one, but they look up and they see that those y values are getting close to two. And the same thing as the limit is approaching zero. As I'm standing on either side of zero, again, I don't know what's happening at zero. That's the function value. When I look up, both people, as I'm getting really close, both people are going, well, it looks like if I kind of follow the graph towards zero, it looks like my y values are also approaching two. Oops. Now one. Well, if I look at negative1 again, the person on just the left side looks up and just the right side looks up. Now, it doesn't look like the y values are approaching two. But you can probably say, well, it looks like they're they're approaching maybe one and a half. So, let's just say 1.5. That's your y value. that is not the function value. So the biggest distinction here is the function value is what's happening at -1. The limits are what's happening around or as you're getting close to -1. And let's erase some of this so we can see better. And then the last one is as x is approaching 12. So let's get rid of these guys too here. Come on. Good boy. as I'm Oops. As I'm approaching 1/2. So, there's What's going on with my pen here? As I'm approaching 1/2, again, I don't know. And I don't care what's happening right at 1/2. I care what's happening just on either side. As I'm approaching, as I'm approaching 1/2 on the x axis, that's what this means. My y value looks like it's heading towards well it's not quite two right it's a little bit bigger than two let's just say I don't know 2.2 or something so it's not the greatest graph that I've given you but the whole idea is you have to make the distinction between what is happening at one or zero or negative 1 or 1/2 which is the function value and what is happening as you approach those numbers on either side. So if you have a graph and are asked to find the limit, it's usually pretty easy. The next way to find the limit is to find it algebraically. So anytime I ask you to find a limit algebraically, and these are going to be some examples of that. This is what you're going to have to do. So, I can't take credit for this little scenario here or this little box of instructions. I actually stole this and this is going to make me sound ancient, but I actually stole this from my high school pre-calc teacher, Mrs. Starky. And uh I this is the first time I learned it and I've loved it and I've taught it ever since. So if you're asked to find the limit algebraically, in other words, you have some ugly function, you don't have a graph, and I ask you to find it algebraically, here are the steps that you have to do. So steps, step one, plug, just simply plug and chug. Plug that number into your function. For example, eight. Your first step, no matter what, is just to plug the number in. So that's going to be 8 over 8 minus 3, which is 8 over five. And while that's not a pretty number, that is just a number. This is our best case scenario because if you get a number out, which we did, that's your limit and we're done. That means that as you are getting close to 8 on the x axis, your y value is approaching or getting close to 8 fths. So that's the best case scenario. So just follow the steps. So there's a lot of things in calculus and I teach this in algebra too. If if there's a certain concept that has set steps like step one do this, step two do this, that should be the easier concepts because you don't have to reinvent anything or come up with anything on your own. The steps are there. You just follow them, right? It's the crazy word problems where you kind of have to think outside the box that are more complicated. So when I ask you to find a limit algebraically, I want you to go, "Yay, I have steps. I don't have to think too hard." Right? All right. So let's do another one. Plug in -2. All right. Now x approaches -2. So I want to find the limit of this function as x approaches -2. First step, well plug the number in. So I'm plugging in -2, which means I'll have -2 all^ squared, which is pos4us 4 all over 0. And now I get 0 over 0. O, that's not good. That's not what we want. If you get zero over zero, so we're going down here. So, this is what we're getting in this one. If you get zero over zero, you've got to try something else. T. Remember this t. And in fact, when you're doing these limits on tests and whatnot, I want you to follow these steps. Plug your number in. And if you get zero over zero, I want you to write down tie. Try something else. Because bottom line, and this is from Mrs. Starky 0 / 0 is never an answer. 0 / 0 is never an answer. So, we're going to have to do something else. In general, there are two algebraic things that will fix this try something else scenario. This is what we call an indeterminate form. It's not done yet and it's unfortunately it's algebra. So, algebra alert. So, this is one of the first times where you know you may this may be fresh in your mind. You may have just taken some algebra or you may not have taken it for a while and you might be like, "Oh, wait. I don't remember this." That's okay. That's why I've given you all those crazy worksheets and I've given you the 130 lecture videos and all that and you have some online or some uh onampus uh algebra review sessions that they're doing. So, there's a lot of places where you can brush up on that. If with when I'm doing these, you realize you might need to brush up a little bit and kind of um clear up the cobwebs. So, if it's not perfectly if it doesn't perfectly ring a bell, no worries. you've got plenty of stuff to review and make sure that you're like I said before consistently reviewing your algebra skills because it will go a long way. So the first thing if you get 0 over zero is to try something else and the first thing is let's see if we can factor and cancel. So right here this is kind of like an algebra review. So for my first for number two here x^2 - 4 over x + 2. Let's see if we can factor and cancel. Well, the top factors by what we call difference of squares in algebra. If you remember that it factors to x - 2, x + 2. And the reason I call it factor and cancel is because in algebra once you factor if any quantities any of those parenthesed factors are the same, you can cross those guys off. So I have factored and I have canceled. Heads factor and cancel, right? So once you have factored and cancelled, you're going to redo your limit. So the limit as x goes to -2 of x^2 - 4 over x + 2. And you're basically going to start over again. Well, what's our first step? Plug in your number. So we're going to plug in -2. Oh, oops. I made a big mistake there. The whole point was to get something better. So the whole point is, let me back up. You plug the number in, you get 0 over zero. Okay, that's never an answer. So you go and you try to clean up the function via algebra so that you've got something a little bit simpler to do and you redo the limit from there. So redoing the limit algebraically just means you try plugging that number back in again. And now our best case scenario, we've now got a number out and that is our limit. If I look at number three, I'll just go down here. We've got the limit as x goes to 3 of 3 - x + 6 over 3 - x. Well, our first step is plug in our number. So that's 3 - 3 + 6 over 3 - 3. And actually, while I'm thinking about it, uh just a little kind of helpful tip for doing these lectures uh with videos, sometimes there's going to be topics where you totally understand it and I may be going off a little bit too long. Throw up the speed on the video. Make it, you know, one and a half times speed. But on the flip side, there may be plenty of times, especially with some of this algebra, where you're like, "Whoa, Megan, slow down a bit." Um even though it'll sound funny, it's a very it's a good little tip. It's a um good studying habit. Put the sound or the speed down to like half speed and even though I'll sound like this and it'll sound funny, it'll give you time in case I'm doing things too fast for you. So, it's kind of a benefit of taking online classes versus in person because you can slow me down anytime you want. So, if I'm doing any of this algebra for the next especially the first few sections of uh this course, um there's a lot of algebra that I will go through kind of quickly. So you might want to slow me down in addition to reviewing those algebra review packets. So here I'm going to have 3 minus 9. Well 9 is 3 and 3 - 3 is 0 and I've got 0 over 0 which is never an answer right Mrs. Starky 0 over 0 is never an answer. Try something else. So we've done an example where we had some like x^2 - 4 that factored. So, we tried the factor and cancel route and it worked. If you have a zero over zero situation and your function has roots in it, that's where you want to multiply by the conjugate. So, let's do this problem. Let's figure out I want to change the color here. Let's try multiplying by the conjugate. Now, multiplying by the conjugate in the world of algebra, it's a fancy name for something not overly crazy difficult. the conjugate of the root thing here. And the root thing in this case is on the top. If it's 3 -x + 6, the conjugate is 3 +x + 6. And we're going to multiply top and bottom by the conjugate. So if you need any help with some roots or conjugates, just go back and look at those reviewed materials and look up any sections on roots and you will see in there conjugates. You can also YouTube things, Conhan Academy, and anytime we're doing algebra and I'm mentioning multiplying by the conjugate, you can easily in this day and age just go on to YouTube, type in algebra conjugates and get a quick video factoring, get a quick video. So, I always try to make sure I tell you what the algebraic concept is. So, you can just jot it down and go review it later. So, that's the conjugate 3 + rootx plus 6 and I multiply both top and bottom by that. Now the the trick here and a little pro tip is take the non-conjugate part of it in this case that's the bottom. So in this case when I say the non-conjugate part the top is where you have the conjugate pairs 3 minus roo 6 3 plus roox sorry 3 - rootx plus 6 3 plus roox plus 6 that we are going to want to foil out. but the bottom which is not conjugates. Leave that as it is because we're actually going to end up doing a little bit of cancelling in a second here. So let's go ahead and foil this out. 3 * 3 is 9. And then I have - x + 6 or sorry plus roox + 6 oh 3. Let me get this right here. So, - 3x + 6 plus I can get this right. Then minus 3x + 6, which is good because those will cancel. And then minus well, here's what makes this problem now work out. Any root times itself is what? That's right, itself. So, I've got minusx + 6 * roox + 6. That is, and I'm go over here for a second. that is minus the entire x + 6. So a x + 6 * a roo x + 6 is an x + 6. But because there's a minus in front, don't forget we are going to have to distribute that negative through. So just be careful there. All right. Well, the whole point of conjugates if you do conjugates right, the middle part of the foiling positive 3 root minus 3 root those are gone. So on the top you've got 9 - x - 6. Well that's 3 - x. And now we are back to yay because remember we're trying something else. We're trying to get this thing simplified to where when I redo my limit and I plug that number back in I don't get 0 over 0. So with conjugates if you do it correctly something should cancel out. So, in other words, what I have now, and I'll try to squeeze this in here, is now I'm going to redo the limit as x goes to 3 of 1 over because I've canceled out the thing on the top. So, there's 1 over 3 + x + 6. And once I've quote unquote tried something else, it should simplify down to where something cancels to where when you plug that number in, it works out. So now if I redo this limit that's going to be 1 over 3 + 3 + 6 again which is 1 over 3 + 9 9 is 3. So I'm going to have 16. Okay. So sorry for that squeezed in there. I didn't give myself enough room. But hopefully you can if you need help you can back it up a bit and look at it again. But I've now got my number out cuz remember if once you get a number out that is your limit. Now the only other scenario and this is another kind of good case scenario is the best case scenario is you plug your number in you get a number out automatically. Boom. You're done. That was our first example. We plugged a number in, we got a number out. Great. Our next two examples unfortunately we plugged our number in and we got 0 over zero which is right. Yep. never an answer. And we had to do some algebra to get it to a point where it simplified to where when we redid it, we got our answer and we got a number out. The other scenario is you might not get 0 over zero. You'll get some other number over zero. So you get like 5 over 0, 6 over 0, 3 over 0. That's what we call an infinite discontinuities discontinuity. And I'll talk about that in a later section. But for right now, all that means is you're good. You're done. There is no limit. The limit does not exist. So if you again following the steps, you take your number, you plug it in. If you get out 0 over zero, ah man, you're going to have to do some algebra. If you get five over zero, all you have to say is limit doesn't exist. So all this stuff over here, it's honestly it's a little convoluted. Um, it's a little bit more fancy than we need to go into, to be honest, because I like to make this stuff as easy as possible without bogging you down with too much notation that isn't really necessary. When there's notation that we really need, I will give it to you. But essentially, all this is saying is you're just plugging in numbers. And it doesn't matter if you just have sorry, one thing inside, maybe you have two things inside, maybe you have a fraction. In fact, we saw that down here with these two, a fraction. All you're doing is just plugging a number in just like here. So, if you were to plug a number in, that's your first step. You get 8 over -4. You get out -2. Yay. You're done. You followed your steps. You plugged a number in. You got a number out. Great. You get 0 over zero. Try something else, i.e. algebra. Any other number over zero limit does not exist. So this is how to find a limit algebraically. So make sure that you note that these are the steps for finding a limit algebraically. Now let's go back to this and we're going to go back to talking about kind of kind of where we started the intuitive definition of a limit. Okay, a limit is not what is happening at two. It's what happening really close to two. In other words, not at two, but what if I was at 1.5 or 1.9 or I got even closer, 1.99, or even closer, 1.999, or even closer, 1.999. As I get closer and closer to two, I'm going to get more of an idea of what the y value or the the the y-value if you want to call it y or the output is approaching. Not what it's what it actually is, but what it's approaching. Now, the next video that I've posted under this is a calculator demo on how to quickly plug numbers in. So, it's a demo on what is called y vars. And if you don't like my demo, go ahead and, you know, you can YouTube it, but just look up y vars for a calculator and it will tell you how to quickly plug numbers into a function so that you don't have to every time go, "All right, 1.5^ squared minus 4, then multiply it by a 4, then remember what that number is and divide it by 1.5 minus 2." Plugging and chugging. So that's what we're doing. So just so you have it written down once. So you're plugging in 1.5 and it's a lot of button pushing, right? It's not necessarily difficult, it's just tedious. So the video on the next page will show you how to do that quickly and it'll save you a ton of time. So if I plug 1.5 in, so what I'm trying to do is find out what is the limit as I'm getting really close to two. So if I plug in say 1.5, you're going to get out 14. If you plug in 1.9, you're going to get out 15.6. 6. If you plug in 1.99, you're going to get 15.96 and then 15.996 and then 15.9996. And you could even if you're bored, you can do five 9.9999 15.9996. Now your answer is not the limit is 15.999996. This is the person who's standing just on either side of two and looking up and going, "Oh, it's at 15.99999999999." You go, "Oh, okay. It looks like it's approaching what exactly 16." So looking at what happens as your x values get closer and closer to two, what happens with your yv values? What does the pattern look like it's approaching? And it looks like at some point, yeah, it's approaching 16. The only time I want you doing this chart is if I literally say, hey, do this, fill out this chart to determine the limit. Okay? So, usually I will say do the um limit algebraically, in which case we've discussed that. So, if I say do it algebraically and you do the chart, that's not exactly what I'm asking you to do. So, just just take note of that. And your homework is the same way. They'll have some where you're doing these charts and then some where you're doing it algebraically. So that said, if I had said, let's do this one, that's the one we just did the chart for algebraically, we're following our steps, right? Our first step is to plug in two. So that's going to be 4 and then 2^ 2 - 4 over 2 - 2. So that's four and then 2^ 2 is 4 and then 2 - 2. And then you can do the math there. You're going to get zero over zero. And then you go, "Ah, dang it, Megan. Now I got more work to do." All right. Well, that means tie 0 over zero is never an answer. Well, I don't have any roots here. So roots is when I want to do the conjugate business. So let's just see if I can factor this. So algebra alert, let's see if we can factor and cancel. Well, t ^2 minus 4. Kind of saw that earlier. That's a nice difference of squares. The bottom is pretty much done. You can't do anything there other than, you know, if you wanted to, you could take out a one. So, the top factors and there we go. That's what you want. You want something to cancel out. So, now we're going to rewrite our limit. The limit as t goes to two of what's left. I didn't cross off the four and I still had that t + 2 left over. So, we start over and we plug in our number. So, let's plug in two. you're going to get 4 * 2 + 2 which is 4 * 4 which is 16. So we get exactly what we got when we did it on the with the chart. Okay? So if I gave you that limit and said, "Hey, fill out this chart." You would see that it's approaching 16. But if I say, "Hey, do that limit algebraically." That's where you plug your number in. You hope you get a number out right away. or you hope you get out, you know, five over zero or 6 over zero or something other than zero over zero because that limit does not exist. And if you get zero over zero, that's where you TSE and you have to do some algebra. But we're not always just approaching a number. Sometimes we may not be approaching two. We may be approaching positive or negative infinity. And if we can find what the limit is as x approaches infinity or negative infinity, we can find what that limit is. That actually is a horizontal asmtopte. And we're going to talk more about horizontal asmtopes, I think, more in chapter 4, but for right now, just keep that in the back of your head that limits approaching infinity are technically horizontal asmtopes. So you you know again you might not be fresh in your brain but you did do a little bit of horizontal and vertical asmtopes in either college algebra or if you took pre-calculus you saw that. So we'll talk more about that later but for right now if you have the limit as x approaches some number let's say you know we'll just put a number this is where we do the steps from Mrs. Starky. Okay, if you have the limit as x goes to infinity or negative infinity, that's where you do this different algebra trick. So, it's very a very common mistake is once I show you how to do this trick on this next page, students tend to like that and then when it's the limit as x approaches 2, they do this same weird algebraic trick. It doesn't work. The only time you're doing this algebraic trick that I'm going to show you here is if x approaches infinity. So in other words, the limit as x goes to any number has its own specific way of doing things that we've discussed. And limits that approach infinity have their also own different way of doing it. But just like before with any other limit, if you have a graph, life is easy. If you don't have a graph, we do this algebraic trick. So graphically, let's look at what we've got going on here. And don't forget our negative infinity and our positive infinity. They're just really big numbers on either end. Okay? So negative 5 billion and positive 5 billion. So if I look at this graph, the limit as x goes to infinity for this graph. Well, as I'm standing over here at 500 bazillion and I do my thing where I look up or down to see where the y-value is approaching. If I'm standing over here, I look up and it looks like that graph, it's not hitting two, but it is what's our word? Approaching two. And likewise, if I'm standing over at negative infinity and I look up or down, I go, "Oh, there's my graph. Oh, yeah. It looks like it's heading." Oh, yeah. It's not hitting two, but it's approaching two. So, that's my limit as x goes to positive or negative infinity. For this next graph, as I'm standing over at 5 billion, I look up and it looks like my graph is heading towards it's not hitting it, but it's getting close to pi / 2. And as I'm standing over here at negative infinity, I look up and I go, "Oh, wait. There's no graph there. What the heck?" And you look down, you go, "Oh, there it is. Where's my function approaching?" Oh, looks like it's getting close to a y value of pi / 2. So these two limits exist because two is a number and pi / 2 is a number. So a little little pre-calculus trig thrown in there for you. But this limit over here will not exist. D ne does not exist because as I'm standing over here so I'm approaching 9 bazillion on the x axis. as I look up, well, my my graph isn't, you know, approaching some asmtope or getting close to a number that's clear, my y-value is going all the way up to the ceiling. So, as my x value is really big, as I'm getting to positive 9 billion, I look up and my y-val is going to positive 9 billion. So, as I'm at positive infinity, my y-value is also going to positive infinity. And the same thing as I approach negative infinity. As I'm getting close to negative infinity, I look up or down. I look up and I go, "Oh, there's my graph. Oh, wait. Where's it going?" And it's it's going shooting all the way up to positive infinity. I will take either answer. You can either say the limit is infinity. Oops, not de. Try that again. Does not exist. Or you can say does not exist. Technically, it's does not exist. The reason it does not exist is let me bring Miss Mick Starky back into this. The reason it doesn't exist is because technically infinity is not a number. Again, quote from Mrs. Starky. Infinity is not a number. So for this particular graph, as I'm approaching positive or negative infinity, I look for where my y values are, they're shooting all the way up to the ceiling. Or if the graph was maybe an upside down parabola, you know, they're approaching negative infinity. Either way, technically it doesn't exist. So, just like before, if you have a graph, easy peasy. If you don't have a graph, a little bit different. So, what we're going to do is talk a little bit about what happens if you have the limit as x goes to infinity or negative infinity and you've got some number on the top and some power of x on the bottom. It could be x^2. It could be, you know, 5x^2. It could be 2 overx to 5th. It could be 11 over x to the 10th. It could be 3 over x to the 7th. Any number over some power, some x power. If you have some number over a power of x, as x is getting super big, your limit is going to go to zero. Well, why? Well, let's just take an example. Let's say I did the limit as x goes to infinity of let's just say 7x^2. Well, you're going, "Well, huh, how do I plug in infinity?" It was one thing when x approached two, how the heck do I plug in two, right? Well, well, how the heck do I plug in infinity? Well, what you can do is you literally can just plug in a bunch of nines because that's what infinity is. So, you can plug in like 9999 squared or something crazy. If you plug in too many nines, it might give you an error. But what you're going to get out is usually 0.00000000 a bunch of zeros depending on how many nines you plugged in. Seven. And what that means is just like when we had 15.99999, we said, "Oh, that's approaching 16." We're not going to say here that, oh, we're approaching 0.7. What we are approaching is zero. Just like 15.9999996 was approaching 16.00007 is approaching zero. So that little algebra trick is going to help us do these problems. And I'm going to do all three of these kind of simultaneously. So, the first thing we're going to do is you're going to look at the denominator and you're going to find what is the biggest power of x that I have. Well, for this one, I've got an x squ. For this one, I've got an x to the 6th. And for this one, I've got an x to the 5th. So, I'm going to divide every term by whatever that is. So, I'm going to take every single term, all of these, and divide it by, in this case, x^2. So 3x^2 /x^2 - x^2 - 2x^2 5 x^2x^2 + 5 x^2 + 4x^2. Now we're not done, but let's just do them all at once. Well, then I'm going to have x 5th over the biggest one in the bottom. So that's x 6 - x3 / x 6 + x / x 6 - 1 / x 6 x 6 / x 6 + x^2 x 6 + 1 over x 6. All right, keep going. Now the biggest one in the bottom, and here's another very common mistake. It's not the biggest x or degree overall. It's the biggest degree or exponent in the bottom. So this is going to be x 6 over x 5th - x 5th + 2x 5th all over x 5th over x 5th - x^2 x 5th + 1 x 5th. All right. Now let's simplify if we can. Well here let's change colors. x squares cancel. So I just have three here. An x cancels. So I've got 1 overx. Nothing cancels on the third one. On the bottom x squares cancel. I get five. One of these x's cancel. So that's 5x. And then I have 4x^2. Let's simplify down here. X 5th cancels leaves me with one there. So that's 1 /x. Three of those cancel which leave me with three there. 1x cubed. X cancels there leaves me with X 5th. Then that last guy can't do anything there. X 6 over X 6 is 1. X^2 X 6. I can cancel out two of them. So that's going to be 1X 4th plus 1X to 6. And finally down here x to the 5th literally now this is the first time the bottom is completely cancelled and I'm just left with one x cancels there leaves me with x to the 4th on the bottom x 5th over x 5th is 1 x^2 of those cancel so I've now got x cubed plus 1 over x 5th all right so we've done our first step we've divided all terms by the biggest x, the biggest degree in the denominator. And now what we're going to do is we're going to use the fact because remember what we're doing. We're doing the limit as x approaches infinity or negative infinity. It doesn't you could change all these to negatives. It's the same idea. Technically, it's not, but for our purposes, it's fine by me. You are taking nine bazillion and plugging nine bazillion into all of these x's. And here's our little algebra trick. When you have some number over a power of x, when you plug in infinity, they go to zero. 1 over a big number goes to zero. 2 over any power of x goes to zero. 5 over any power of x goes to zero. 4 over any power of x goes to zero. This goes to zero. One over billion numbers cubed. 0 1 over billion to the 5th. 0 1 over billion to the 6th 0. 1 over x 4th or billion to the 4th. 0 1 over x 6 0 1 over notice I'm skipping this. That's not one over a power of x. That's just x. Two over a billion x to the 5th. Zero. One over a bazillion cubed. Zero. one over a bazillion to the 5th zero. And if you look at what you have left for this first one, you've got three minus a bunch of zeros, five plus a bunch of zeros. Hey, we got our number out. Sweet. We're done. Here on the top, I've got 0 - 0 plus 0 minus 0, otherwise known as zero. And on the bottom, 1 plus 0 plus 0, otherwise known as one and 0 over one. Common mistake. People think it's like undefined. You just can't have a zero in the bottom. Okay? There's some cheesy thing called the ten commandments of math and it says, "Thou shalt not divide by zero." We're not dividing by zero. We're divided by one. And honestly, if you're not sure, take out your calculator. 0 divided by one is zero. That is a legitimate answer. Sometimes students think, "Oh, zero, something must be wrong." Zero is a number. Remember our rule with limits? You want to end up getting out a number. Zero is a number. It's just that zero over zero that we have to be careful of. And then here's the tricky bit. We still have this x. Only a little mini fraction with a power of x in the bottom goes to zero. So all this stuff in pink is zero. But that still means I have x over one. In other words, I still have x. And the limit as x gets super big of x. Well, if I'm plugging in, think of our first rule of of of limits. You're plugging the number in. If I plug in 9 bazillion just in for x, I get out 9 bazillion, otherwise known as infinity. or as we saw above, if your answer is infinity, technically that means the limit doesn't exist. So the most important thing here is to distinguish that when you have a limit as x approaches anything other than infinity, you're following that set of TSE steps. If you have a limit approaching infinity, you are doing well, unless you have a graph, you're doing this crazy algebra trick. And then the last thing in this particular section is to start looking a little bit more at limits from graphs. So I've given you three graphs here and then we've got one graph that is not done and we're going to graph it ourselves. Now this is what's called this well all of them are but this one in particular is what we call a peiewise function. And so I will show you this graph or how to graph this um just from an algebraic review standpoint. Like I said, you know, as needed, we'll revisit some algebra, but if you're having trouble, um whether it's with this one or the ones you see in the homework, um definitely go back to those um not only as I say here, page 59 in our book, numbers 49 through 47 through 50. There's some more examples there. But just go ahead and head into that algebra review worksheet page in Canvas, and hunt down, in fact, in there, there's a very specific one-page full step-by-step algebraic review on how to graph peacewise functions. So, you might want to take a look at that. But if you already have the things graphed, we're good to go. So what we're going to do is we want to find the limit based on graphs we already have. Here we want to find the limit as we are heading towards -2. Here we want to find the limit as we're approaching three. And here we want to find the limit as we're approaching zero. So here this is basically what we saw before. If I'm standing right on the right side or just on the left side of two, both people look up. And again, we don't care what is happening at -2. We care what's happening around it. Right on either side of -2, both people look up and they go, "Oh, it looks like the y-value is approaching three." There you go. Now we're going to start to talk a little bit more about versus this is the limit versus the function value. The function value, recall from algebra, the function value is what is actually happening at -2. What is the y value at -2? And if I look up, go right to -2 and put a big old blue dot, it's definitely three. f of -2 my function value is also three. So f of -2 is representative of the function value. The fancy way of saying what is happening at -2. So at -2 the y-value is three and actually around -2 it looks like it's approaching three. In comparison here, if I'm looking at the limit as we approach three, if I'm standing on either side of three and for a temporary minute, I don't care what's happening there. So basically, the thing in yellow is the function value and everything around it is the limit. So the person over here looks up, they have no idea that there's a hole there. Okay? They can't see anything but what they see right above their head. Both people look up on either side and they go, "Oh, yeah, looks like the y-v value is three." Okay, that is what the limit is on either side of three. On 2.99 and 3.01, they look up and it looks like the y-val is approaching three. However, the function value is what is happening at three. and at three open circle in algebraic terms means that's actually not the function value. That is not the y-value. And where do we find a function value on a peiewise function? Yep, that's right. If there's a hole there, that means there's no function value there. But if we have a little dot up there, it's saying, hey, that's where the y-value is. So on either side of three, it looks like the function is heading towards three, and that's the limit. That's correct. But the function value itself is four. For this one, well, if I'm want to know the limit as we approach zero, here we have our first issue of what are called onesided limits. One-sided limits. So, I keep saying and I've said from the beginning, you want to look at what's happening just on the left side and what's happening just on the right side. And up until now, even our first graph that we did, the left and right side were all approaching the same y-value. But here, the person on the left side, they look up and they go, "Oh, look at my function is right there. Looks to me like it's heading towards one." Okay, they they're standing right there and as far as they know, they look up and they see that thing heading towards one versus the person on this side, they go, I don't know what you're looking at, but I look up and my graph is going all the way to infinity. So each side of the one-sided limits, which we'll talk about in the next section, is telling us something different. The dude on the left looks up and sees the function headed towards one, and the dude on the right looks up and sees the function going all the way to infinity or not existing. So the question is well what what the heck is the limit and we'll get to that in the next section. So like always if the graph is given to you life is easy. If it's not given to you then you're going to have to graph it. So let's quickly do this and I'll explain a little bit of the algebra on how to graph this. And again you you may want to go back if it's a little shaky. Peacewise functions can be shaky for a lot of students. So just go on back and and and do a little bit of review um on that particular concept concept. So what we have we're going to graph and the way we want to graph peacewise functions is we want to graph each piece at a time. So we want to graph that that and that separately. And in fact I'm going to do this one. Then I'm going to do this one. And then we'll save the middle one for last. What used to get me what confused me when I was learning this stuff in algebra is this side. That's secondary. Don't think about that too much. So the first thing we want to do is just graph each piece. And remember f ofx and y are interchangeable. So sometimes it's helpful to look at this as y equals. So our first piece is the line y = x. Our next piece is the line y = 0. And our last piece is the line y =x + 2. This piece and this piece are just mx plus b, right? Just regular straight lines. And if you want to write that in there, y = m, which is 1. x + b. For our last piece, y = -1x + b. So, the first and last pieces are just nice straight lines. And you'll have some in the homework and you'll see some in the review sessions where you might have some parabas. You might have to review where parabas shift up and down, left and right. But the bottom line is with peacewise functions, you just want to take each piece and graph them separately. Just go, what would I do if I was just asked to graph 1 x + 0? Well, that's a y intercept of zero. So start at zero and you want to go up one and over one. And so that line all on its own would look something like this. Now one of the key things that I've learned and in um that in the algebra review packets that I'm mentioning um one of the things is um um I can think is a like I said earlier a full page review on how to graph peacewise function from scratch. Um they mention in that finding what I call break points and finding the break points was what made graphing peace-wise function in this confusing part for me so much easier. Break points are simply you take whatever this number is in this case it's one and you plug it into that piece and literally if you plug one into x you get out one. So when one goes in, one pops out. That means one comma one is your break point. Make that break point strong. And then what you're going to do is you are going to keep everything to the left. That's where the little less than and greater than come in handy. You're going to keep everything to the left of one. So, you just look at that dot and you go, "Well, that means I'm going to keep all of this stuff and I'm going to get rid of anything on the right." There's your first piece. Now, the last thing that you need to do Oh, oops. That's not what I wanted to do is determine if that break point is opened or closed. Well, recall strictly less than is an open circle. So, a little bit recall less than or greater than is an open circle. Less than or equal to, greater than or equal to are closed circles. So, since it is strictly less than, we want to take that break point and make it an open circle. Now, what I want to do is graph this. Well, this is just a straight line with a y intercept of two and a slope of 1. So, I'm going to go up to two. Oops, I keep hitting that wrong thing. Slope of sorry, y intercept of two. And now my slope is ne 1. So, I want to go down one over one. And looks like we're going to have to fill that circle in there for a second. There is that entire line. Okay, there's that entire line. Oops, did it again. Oh, wow. I really messed that up, didn't I? Let's not do that. Let's go back. So, now let's find our break point. So, my break point, remember, you take whatever that number is and you plug it in. So, it's going to be 1, well, if I plug one into that, that's -1 + 2, which is positive 1. So here we have a scenario where both of our break points are the same. If your break points are the same, all that means is that those two pieces need to connect. So a lot of times what used to get me is, you know, I might graph something and I might have it look like this and I look in the answers and they ended up being connected. And that used to always frustrate me. Once I learned the breakpoint method, I never had that problem again. So, one of the main reasons to do the breakpoint is so you know, hey, all right, these two lines are going to break at that point. Yeah, but they're in the same place. So, our lines are going to connect. So, we do our thing. We make a real prominent break point there. Okay, we're talking about our blue line now. And this tells us what we're keeping. We're keeping the stuff bigger than one or bigger than our break point. So if I want to keep everything greater than that's all this stuff over here, we're keeping the part to the right of that break point and we are getting rid of that. Well, what's our next step? Is it opened or closed? Hm. All right. Well, our break point here and our break point here are in the same place. However, we're in luck because both of them are strictly greater or strictly less than. So, we are going to have an open circle at 1 one. So, our break points are in the same spot. So, our pieces should connect. But since they're both opened, our break our connection point is going to be open. If they were both closed, our connection point would be closed. Little helpful tip. If one of them is open and one of them is closed, remember this closed beats open. So sometimes people would be like, I don't know, is it like half open, half closed? Is it a close in the middle and like a bullseye or like what is it? So if let's say this was strictly greater than or less than and that was a closed circle, but this one was open. But you're going, hey, they're in the same spot. You know what do I do? Close beats open. So, if they're both open, it's open. If they're both closed, they're closed. If one's open and closed, and they're in the same spot, it's going to be closed. And now, the last piece here is the line y equals 0. Well, y equaling 0. Remember, y equals 0 is a horizontal line through zero. y equals 0 is just a horizontal line through zero. y = 1 would be a horizontal line through 1. y equals a number is a horizontal line. And for purposes that you might need later, just as a review, x equals a number would be a vertical line through that number. So that's going to be kind of weird. So we're saying, okay, our next piece, the middle one, is a horizontal line through zero. It's like, wait, what the heck? And not only that, we don't have a less than or greater than here. We just have an equals. What the hell does that mean? Well, what that means is yeah, the entire line is the horizontal line through zero. But I literally only want to keep the point, not stuff to the left or stuff to the right. Literally want to keep the point that is at one. So yeah, you have the entire line, but you literally only want to keep the point at xals 1. So that's the tricky part of this one. So it's not like it's x greater than one and you would keep just that side of it or x less than one and you'd keep that side of it. It's literally just that point. So if this right tricky side has an equals, that just means that one particular point is what you want. So, peacewise, again, not easy, but you've got a lot of stuff to review off of and a lot of um a lot of stuff that I've provided, a lot of stuff that you can find online to help you refresh your memory on that stuff. But let's not forget the calculus. So, perfect way to end this this lesson. What did I say at the beginning? One thing I've learned after teaching this course for years is it's usually not the calculus. What we've done here, this is just algebra. So what I have done for this problem has zero to do with calculus. Believe it or not, the only thing I'm going to do now is find the limit as x a but x approaches one of this function that I've graphed. So the calculus that I'm about to do is the super freaking easy part. It was the graphing that's the problem. So that's why you really want to review these on your own. really refresh your memory on how to graph these things because if you can't graph them then you won't be able to do the easy part which is the calculus which is the limits right so the limit as I'm approaching one well there's one and if I'm standing just to the left or just to the right in either case the person looking up goes oh looks like the function is heading towards a yvalue also of one. I don't care. Remember, I don't care what's happening right through there. That is the function value. Okay? All I care about is what is happening literally right there and right there. And both of those y values are heading towards one and they're approaching one. And that is the limit. That's the calculus. But and let's full circle this, the function value, what is happening at one. So when you're standing right on one, in this case, you're literally standing right on that point. The function value, where your closed dot is, as we talked about that before, is actually zero. So the function value is zero, but the limit is one, which is similar to this. The function value was up at four, but what was happening around it was actually three. So the limit and the function value weren't the same thing. And when the limit and the function value are not the same thing, what happens is we have what's called a discontinuity. There's going to be some sort of break in the graph. See how there's like a break because it's a open circle and then it kind of there's another point somewhere else. This is another discontinuity. There's a break in the graph. Here there's a break in the graph. Whereas here there's no break. So other than this one, all the three others on this page, the two that were already there and the one that we graphed are examples of having discontinuities because there's breaks in the graph. And what that typically means, and this is one example, your function value and your limits are not the same. So that's where we get into some theory with algebra or with algebra with calculus. So I can't stress enough um you know, my role here is to really is to guide you through this new material. Um, and as you've seen, you know, I'll revisit I'll I'll slow down on a couple things and give you examples of some algebra just to refresh your memory, but it's really impossible for me to to refresh everything algebraically and do the calculus. So, just make sure that you're keeping up with um, you know, consistently reviewing your algebra skills. Um, and um, yeah, like I said, there's a lot of lot of things out there for you to do that with. And I don't know if that sentence makes sense, but we're just over an hour. I thought I was going to try to keep it under. Almost got there. So, that's it for now. And I will see you all in the next section.