 In the study of calculus, this idea of error and tolerance that we've been talking about in the previous videos, these epsilon deltas, is closely related to the concept of a limit, which we're gonna explain in this video. So let F be a function defined on some open interval. Open interval means that we don't include the endpoints, but we have all numbers, say from B until C, which we might write something like this. This is an open interval, which we get all numbers X that are between B and C. That's what we're talking about right here. The domain could be much bigger than that. All we're saying is that there's at least some open interval, and that open interval is gonna contain the number A. But we do allow for the possibility that the function is undefined at the number A itself. So A actually could be missing from the domain. It could be in the domain. We're not making any assumptions. We're just saying that there are points near A that belong to the domain of the function, okay? So then we say that the limit, the limit of F of X as X approaches A is L. And we write this in this compact notation right here that you see in this green box. LIM is short for limit. The arrow on the bottom tells us the limit as X approaches A of F of X is equal to L. What does that mean, right? So we say the limit of F is L if for every epsilon greater than zero, there is a number delta greater than zero such that when the absolute value of X minus A is greater than zero, but less than delta, that'll imply that the absolute value of F of X minus L is less than epsilon. Oh boy, that was a mouthful. This is why in the previous videos we've been putting so much attention to this epsilon delta so that when we get to this moment right here, we can actually have some hope of interpreting what this thing means. So let's try to unravel this again one more time in this video. When you take this expression, the absolute value of F of X minus L is less than epsilon. Well, if you remove the absolute value, you get something like the following. This would say that F of X minus L is gonna be less than epsilon, but greater than negative epsilon. Remember, when you take the absolute value of something, you're forgetting the sign. The absolute value of a positive is a positive. If the absolute value of a negative is still a positive. And therefore, when you take the absolute value of something, you forget its sign. If you wanna remove the absolute value, you have to consider both signs plus and minus, unless you get these two possibilities. The absolute value of F of X minus L, that's gonna, if that's absolute value is less than epsilon, that means F of X minus L is between epsilon and negative epsilon. Then adding L to both sides, in this case, I guess there's three sides, right? Then you didn't get the statement that were indicated here with the arrow. So this statement right here, the absolute value of F of X minus L is less than epsilon. That's just a more compact way of saying F of X is between L minus epsilon and L plus epsilon, which here, this interval L minus epsilon and L plus epsilon, this is our margin of error. As we've drawn in previous pictures, right? We have this strip right here. We have L plus epsilon, you have L minus epsilon, and then L itself is some value in the middle. As long as we're inside of this region, we're inside the margin of error. So this green box you see here on the screen, it just disappeared. It's now a yellow box. This just means the absolute value of F of X minus L is less than epsilon. This just means you're in the margin of error, okay? It's just mathematical notation to tell you that your function's in the margin of error. Now this margin of error is dependent upon epsilon, right? You have to decide how much error do you allow? Do you allow an error of 10%, 1%, 0.1%? And so that's why we say for any epsilon greater than zero, we're in the margin of error. So what we're saying is we're not specifying the level of error. I'll come back to that in just a moment. Let's consider this statement right here. Zero is less than the absolute value of X minus A, less than delta right there. Well, if we ignore this part right here, this unravels in the same way. Saying the absolute value of X minus A is less than delta is equivalent to saying that X is greater than A minus delta but less than A plus delta. So this right here is going to be our domain, our domain of allowance. And so this box right here, right here, this is the domain of allowance. This is, we're saying we're in the domain of allowance. We can tolerate any number inside of there. Because if you're in the domain of allowance, that all imply that the output, if your input isn't the domain of allowance, your output will be in the margin of error. What is this zero less than implying here? Well, this comes back to the fact, remember that our function is not necessarily defined at X equals A. It could be, but it might not be. We don't actually know. And so we actually are saying that, okay, ignore X equals A, right? Because if X equals A, then X minus A in that case would be zero, its absolute value would be zero. It would be quality to zero. So this zero less than statement is just to telling us that we are ignoring the possibility that A could equal X. That possibility we're ignoring right here. Okay, so with that recap there, let's try this one more time. We say that the limit of F of X as X approaches A is L if when any amount of error is prescribed to us, we can always find some antidote of allowance that'll guarantee that if you're in the allowance, the inputs and the allowance and the output will be in the margin of error. Say that one more time. Given prescribed any amount of error, doesn't matter how restrictive your error is, if you give me any tolerance of error, I can always find some allowable domain that guarantees you land inside the margin of error. If that happens, if any prescribed error can lead to this implication here in blue, then we say that the limit of the function is L in that situation. And so I want to show you an example of how one actually can compute a limit using the precise definition of the limit here. So in the remainder of this video, you're gonna prove the statement. The limit as X approaches three of the function four X minus five is equal to seven. So what that means is that if you give me any amount of error, right? So you're gonna give me some error. It doesn't matter what it is, right? As long as it's positive, right? If you give me any amount of error, I can give you a delta that'll guarantee that an input no more than delta units away from the target will land you no more than epsilon units away from the target and the output, right? Now, the concern of course is that if epsilon is greater than zero, that's the only thing we know about epsilon. That's the only prescription that it's positive. That means my delta actually needs to be a formula. I should say my epsilon, there needs to be a formula that relates epsilon and delta together so that if you gave me a specific epsilon then delta would be a formula with respect to that epsilon. So how might one do that? Well, it depends a lot on the function, right? And that's something we're gonna discover over the next several lectures, that functions with regard to the limits can be behaving very differently from each other. So what we do know is, well, what we want, this is what we want right here. It could be that it's impossible. There might be some epsons which don't have a delta. That means the limit might not exist in that situation. So if this is the situation we want, what can we say with that? Well, let's play around with the right-hand side. If we take the absolute value of 4x minus 5 minus 7, well, we could combine like terms, that's gonna be the absolute value of 4x minus 12. I noticed that 4x and 12 are both divisible by four, whose absolute value is also four. If you factor it out, you get four times the absolute value of x minus three. But hey, wait a second, I know something about the x minus three, right? If you look at x minus three, that's supposed to be less than delta right here. So we know that absolute value of three is gonna be less than delta. But we also know that four times the absolute value of x minus three is supposed to be less than epsilon. If we divide both sides by four, we're gonna get the absolute value of x minus three is less than epsilon over four. And so that actually gives us the connection we want. We're gonna, if we set delta to be epsilon over four, then that turns out that's gonna work for us. Where the significance of four is after all this function y equals four x minus five, this is a linear function, four is the slope. And so in this situation, we use the slope to determine the ratio between epsilon and delta. For linear functions, epsilon and delta will be directly proportional to each other with respect to the slope. Now what we saw here is technically not a proof. That's just one, this is sort of like the thinking that one could use to derive the proof. Here's the actual formal proof that I can prove that the limit of four x minus five equals seven as x approaches three. So the proof looks like the following. Let epsilon be greater than zero. So I don't know what epsilon is, but I can still choose delta. I'm gonna choose delta to equal epsilon over four. So suppose the absolute value of x minus three is less than delta, but greater than zero. So that is the distance between x and three is no more than delta, but x does not equal three. Then in that situation, consider the absolute value of four x minus five minus seven. Simplifying this, this becomes four x minus 12, which becomes four times the absolute value of x minus three, which by assumption x minus three is less than delta, we're gonna get that this is equal to, this is less than I should say, less than four times x minus, excuse me, four times delta, which delta is epsilon over four, in which case four over four cancels, you get epsilon right here. And so therefore this choice of delta, no matter which epsilon you give me, I can always just divide it by four, and that'll give me a delta that guarantees that when you are delta close, then you will be epsilon close. Therefore seven is the limit. Now this example right here, I'll be honest, is probably quite overwhelming for the typical calculus student. If I give a little bit of backstory, I took calculus in high school, and then I elected to repeat it in college as well. So I actually took calculus one twice as a student. And therefore I was exposed to these epsilon delta proofs twice, both in high school and in college. And you know what? The first time I saw it in high school, didn't get it. The second time when I saw it in college, guess what happened that time? I still didn't get it. It wasn't until many semesters later, and actually what's called real analysis, which is a class about proving stuff about calculus. It wasn't until then when I started writing my own epsilon delta proofs that I truly started to understand what the importance of delta and epsilon are in this type of setting. And so if you are struggling right now, that is okay. You are welcome. You have friends. You're not alone in that struggle. And that's okay. Now the good news for us is that while this precise definition of the limit is important, especially from the theoretical point of view, and anything we actually know about limits will be proven using this definition. The good news for many of us as we proceed forward in this class is that we will rely more on an intuitive notion of what a limit is, and use that to help us with calculations. And that intuitive limit is what we're gonna talk about in our next video of this lecture series.