 In our previous video in lecture 10, we introduced nine limit laws that help us to algebraically compute limits of functions. And this covers, those properties cover the vast, you know, diversity of types of elementary functions one sees in precalculus and in calculus. And so that, those limit laws do really, really great. But one of the limit laws I wanna point out that we saw earlier was it was number D that told us that the limit as X approaches A of F of X over G of X, this is equal to the limit of F of X over the limit of G of X, of course as X approaches A in both situations. This is just, so the limit of a quotient is a quotient of limit, but the one exception would be when this bottom was zero, right? So this limit law only applies if the denominator is non-zero, but if the denominator is zero, turns out funky things can happen. The limit might exist, it might not, it could be a vertical asymptote. There's some interesting things going on there. So we have to investigate it a little bit further. And so to help us in that regard, we're gonna add our 10th limit law, number J here, and which tells us that if two functions F and G, if they agree on their domains, except maybe at X equals A, of course if they agree at A then they would be the same function, but maybe two functions disagree at X equals A, but they agree everywhere else. Then that means that the limit of F is gonna equal the limit of G as X approaches A, assuming that these limits exist at all. So if two functions only disagree by a single point, then their limits will agree on that same point. Let me give you an example of such a thing. So suppose we were to have some function, here's our X axis, here's a Y axis, and we graph our function, that's maybe it's doing something like this, do, do, do, do, do, right? This is a nice, pretty function for which when you look at the graph, if you pick any point, any point on the graph right here like this one, when you start approaching the point from the right, you approach it from the left, you're gonna get, oh, it's this point right here, the limit would exist in that situation. But let's say that we modify the function in one single way. What if we change what happens at this point to just be poof, it's now gone? That is to say, what if we remove the point, okay? So G is the same function as F, except at this point, let's say it's X equals A at this moment, X equals A, at X equals A, the function disagree, the function F disagrees with G because for F, let's say it was defined to be something like three comma two, right? In that case, the limit as X approaches three of F of X in that situation would have been two. Now, in fact, we saw that F of X is equal, sorry, F of three is in fact equal to two in that situation. But with this other function, when you look at G of three, that does not exist. The function's not defined at three, but there's this remove point. But because G and F are the same function, except for what's happened at three, we see that the limit as X approaches three of G of X is likewise gonna be two. The expectation of what G ought to be doing is still two, even though it's defined to be something else. Or as a third possibility, right? Let's say we have some other function H, which is defined, right? What if H is defined here? You have this point three comma five or something like this. In this situation, H of three is equal to five. That's how the function is defined. But the limit as X approaches three of H of X here, this is still going to be two. All three of these functions, F, G and H, are the exact same function except at X equals three right here. And because the functions agree, they agree around X equals three. They don't agree at three, that's fine. But when it comes to computing the limit, you don't actually care what happens at three. You care what happens around three. And these three functions, F, G and H, they all agree around three. Therefore the limit as you approach three will be the same. Let me show you how this can be used practically. So consider the function F of X equals X minus one over X squared minus one and determine whether there's a vertical asymptote at X equals one or what's going on at X equals one. Now, just to be clear, if you're looking for a vertical asymptote at X equals one, that means you're trying to determine the limit as X approaches one. And you can approach it from the left or from the right, it doesn't matter of F of X here. If that turns out to be plus or minus infinity, if the left-handed limit or the right-handed limit turns out to be infinity or negative infinity, right? If one of those limits is infinite, that means our function has a vertical asymptote. So we're trying to compute the limit here of F of X as take compute the limit of F of X here when you get of X minus one over X squared minus one. Now you can't just plug in X equals one, right? Because if you tried that, you're gonna get stuck, right? F of X, sorry, F of one in this situation, you get one minus one over one squared minus one, which gives you zero over zero. So division by zero means like, oh, evaluation here doesn't work. We can't just plug in X equals one to determine the limit. But when you see something like this, zero over zero, I want you to think of like Princess Leia right here because it turns out zero over zero is there's hope in that statement. Turns out this limit could be something else. I mean, at the moment, we don't know what it is. We need to find out more. This is what you call an indeterminate form. That is, from this form, we cannot yet determine what the limit is. The limit might not exist. It could exist. It could be finite. It could be infinite. We don't know. We have to investigate a little bit further. And so this actually comes down to an algebraic thing. If we look at F of X here, if we factor the denominator, you'll notice that X squared minus one is a difference of squares. You get X minus one and X plus one for which in a previous algebra setting, you probably were taught, hey, X minus one over X minus one, those cancel each other and you end up with one over X plus one. But now this is one of the most fundamental lies of all of mathematics right here. These statements right here are not actually equal to each other. These are not the same function because if I take F of one, for example, dude, this is equal to zero over zero, which is not a number. On the other hand, this one right here, if you plugged in one, you would end up with one over one plus one, which is equal to one half. Is one half equal to zero over zero? Well, zero over zero is not a number and one half is. So in all reality, these functions are not equivalent to each other. So when we say things are equal, we mean that it's equal on their common domain. So it's equal so long as X doesn't equal one because the function on the left is undefined, the function on the right is still defined. So on their common domains, the original function F, its domain is all real numbers except for one and negative one. This simplified function, this function in lowest terms, its domain is actually all real numbers except for negative one, it's well-defined at one. So these functions do disagree with each other. F of X, its domain here, right? Its domain, like I said, this can be all real numbers X, except for X is plus or minus one. This other function, let's call it here G of X, we see that the domain of G is equal to all real numbers X, such that X doesn't equal negative one. G of X is defined at one, even though F of X isn't. And that's the only difference between F of X and G of X. F of X will equal G of X for all real numbers except for at positive one, for which at positive one, F is undefined and G is defined. This is the setting we were thinking about right here, right, where F and G disagree with each other only at one point, maybe because F was undefined, but G in this case is defined. Therefore, the limit of F of X here is gonna equal the limit of G of X. So we see that the limit as X approaches one for this function here of X minus one over X squared minus one, this will equal the limit of the simplification. X approaches one of one over X plus one, like so, in which case then we see that this limit's gonna equal one half. And so since the limit is equal to one half, we don't have a vertical asymptote on, we don't have a vertical asymptote for the function F at one, we actually have what's called a remove point because this limit did not go towards positive or negative infinity, turned out to be a real number. So this would be like somewhere on the graph, we have that missing piece. So you just imagine Golem is now screaming, they stole it from us, they so installed this point. And we can see that because the function F and G, since they differ just by the one point, their limits will be the same. And the limit of G wasn't infinity, it was actually one half.