 Alright, let's talk about one of the stranger ideas in mathematics, which is the notion of a limit of and or at infinity. And so this concerns the meaning of this peculiar symbol here, this figure eight, or if you want to show your erudition, which means you know a lot of words, we might call this a lemniscate. And this occurs in the following way. Many quantities that we deal with in the real world have a bound. So for example, the number of hours you can study in a day has an upper bound of 24 hours. It is impossible for you to study more than 24 hours in one day. This also is a quantity that also happens to have a lower bound of zero hours. You cannot study less than zero hours in one day. And so the amount of hours that you actually spend studying for a course, say calculus, is going to be somewhere between the two bounds. Now it's unlikely that it's going to be anywhere close to this 24 hours because that 24 hours would also include times that you're eating, sleeping, going to other classes, and doing other things. But one also hopes that it's going to be somewhere away from this lower bound of zero hours. Reasonable amount of time to study calculus in one day, I don't know what, 16 hours in a day is a reasonable amount? Maybe, probably not, but there is some value for the number of hours you spend studying calculus. Now those upper and lower bounds don't have to be set by some real physical constraint. It's impossible for a day to have more than 24 hours and less than zero hours. It could also be just set by some sort of arbitrary limit. So for example, if you're a full-time student, typically the number of credits you can take also has an upper bound, usually around 18 credits or so. And a lower bound, again, typically around six credits. If you drop below six credits, you are no longer considered a full-time student. So if you are a full-time student, you have both an upper bound and a lower bound for the number of credits that you can take. On the other hand, many quantities do not have bounds, that these quantities are unbounded. So for example, if I consider the function f of x equals x squared, there is no upper bound on this function. I can make x squared as large as I want by picking a sufficiently large value of x. So there is no upper bound, however, this does actually have a lower bound. x squared cannot be less than zero. No value of x that I can substitute into this function. No real number will have a square that is less than zero. So here's a function with no upper bound, but with a lower bound. On the other hand, I can take a function, nice, simple tame function, like 8 plus x, and here there's no upper bound and there's no lower bound. I can make 8 plus x as large as I want by picking a large value of x. I can also make it as small as I want by picking a small value of x. So to express the idea that a function or an expression has no upper bound, we use the symbol plus infinity. And likewise, if I want to express the idea that there is no lower bound, I'm going to express that using the symbol minus infinity. So let's take a look at a problem. We'll start off with limits at infinity. So I want to find the limit as x goes to infinity of 1 over x. What happens to 1 over x as x goes to infinity as x gets larger and larger without having a bound? And again, we'll try and defend our answer by looking at a numerical approach. So again, the expression x to infinity means that x is increasing without bound. And I'm going to look for an answer numerically in any case, so I might as well start by considering a table of values. So I'll consider some values of x and some corresponding values of 1 over x. Since x is increasing without bound, then I probably want to take a look at large values of x and make them even larger. So let's start off with a large value of x. So let's think of a big number. How about 10? Well, that's not really a big number. How about a thousand? So we'll try x equal to a thousand. And substitute that in 1 over x, 1 over 1,000.001. Now the important thing to note here is we are considering what happens as x goes to infinity as x continues to get larger and larger. So if my first value is 1,000, I want to think of a larger value of x. And so how about 1,001? Well, that is larger. But let's take a giant leap. Let's go all the way up to 10,000. And let's take a look 1 over x, 1 over 10,000.001. And by now you should get the hang of it. If I want to get a larger value of x, well, let's try 100,000 and see what happens 1 over x looks like .0001. Now it might not be entirely clear what's happening to 1 over x. So here's where a little bit of algebra will become useful. As x gets larger and larger as x goes to infinity, then this expression 1 over x, what I'm doing is I'm taking a very big number and using it as a denominator. And if I have a big number in the denominator, my fraction is going to become a small number. And so it looks like 1 over x is heading towards zero. And that allows me to make the conclusion the limit as x goes to infinity of 1 over x is going to be zero. Now it turns out this is actually a very general rule. In general if I have n greater than zero, if I have my exponent as positive, then in general as x goes to infinity, the expression 1 over x to the n is going to tend to zero. Now because we are taking the limit as x goes to infinity, we refer to this as a limit at infinity. And a similar sort of argument would hold if x goes to negative infinity. But what about other types of limits? So for example let's consider the following. We are going to take a look at the limit of 1 over x as x gets close to zero, but always staying a little bit above zero. And in this case we again want to support our answer numerically. So our notation here, x is getting close to zero, but always staying a little bit more than it. Since we do eventually want to take a look at the numerical result, let's go ahead and start with that. So again I'll take a look at my values of x and values of 1 over x, and I'll take a value of x close to zero, but a little bit more. And if x is 0.1, then 1 over x, 1 over 0.1 is 10. If I get a little closer to zero, my 1 over x value, 1 over 0.01, 100. And if I get even closer to zero, 1 over 0.001, that's going to be a thousand. And again it might not be entirely clear what's happening to 1 over x as x gets close to zero, but we might make the following algebraic argument. This expression, 1 over x, if x is getting close to zero, if x is a small number, if x is a small number, then 1 over x is going to be a large number. And so as x gets close to zero, 1 over x looks like it's going towards larger and larger, and we say that it's going to go to positive infinity. And so our limit as x approaches zero from above of 1 over x looks like it's going to go to plus infinity.