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AP Calc: LIM‑1 (EU), LIM‑1.C (LO), LIM‑1.C.1 (EK), LIM‑1.C.2 (EK), LIM‑1.C.3 (EK), LIM‑1.C.4 (EK)

Video transcript

- [Instructor] So we have the graph of y is equal to g of x right over here. And I wanna think about what is the limit as x approaches five of g of x? Well we've done this multiple times. Let's think about what g of x approaches as x approaches five from the left. g of x is approaching negative six. As x approaches five from the right, g of x looks like it's approaching negative six. So a reasonable estimate based on looking at this graph is that as x approaches five, g of x is approaching negative six. And it's worth noting that that's not what g of five is. g of five is a different value. But the whole point of this video is to appreciate all that a limit does. A limit only describes the behavior of a function as it approaches a point. It doesn't tell us exactly what's happening at that point, what g of five is, and it doesn't tell us much about the rest of the function, about the rest of the graph. For example, I could construct many different functions for which the limit as x approaches five is equal to negative six, and they would look very different from g of x. For example, I could say the limit of f of x as x approaches five is equal to negative six, and I can construct an f of x that does this that looks very different than g of x. In fact if you're up for it, pause this video and see if you can so the same, if you have some graph paper, or even just sketch it. Well the key thing is that the behavior of the function as x approaches five from both sides, from the left and the right, it has to be approaching negative six. So for example, a function that looks like this, so let me draw f of x, an f of x that looks like this, and is even defined right over there, and then does something like this. That would work. As we approach from the left, we're approaching negative six. As we approach from the right, we approaching negative six. You could have a function like this, let's say the limit, let's call it h of x, as x approaches five is equal to negative six. You could have a function like this, maybe it's defined up to there, then it's you have a circle there, and then it keeps going. Maybe it's not defined at all for any of these values, and then maybe down here it is defined for all x values greater than or equal to four and it just goes right through negative six. So notice, all of these, all of these functions as x approaches five, they all have the limit defined and it's equal to negative six, but these functions all look very very very different. Now another thing to appreciate is for a given function, and let me delete these. Oftentimes we're asked to find the limits as x approaches some type of an interesting value. So for example, x approaches five, five is interesting right over here because we have this point discontinuity. But you could take the limit on an infinite number of points for this function right over here. You could say the limit of g of x as x approaches, not x equals, as x approaches, one, what would that be? Pause the video and try to figure it out. Well let's see, as x approaches one from the left-hand side, it looks like we are approaching this value here. And as x approaches one from the right-hand side, it looks like we are approaching that value there. So that would be equal to g of one. That is equal to g of one based on that would be a reasonable, that's a reasonable conclusion to make looking at this graph. And if we were to estimate that g of one is, looks like it's approximately negative 5.1 or 5.2, negative 5.1. We could find the limit of g of x as x approaches pi. So pi is right around there. As x approaches pi from the left, we're approaching that value which just looks actually pretty close to the one we just thought about. As we approach from the right, we're approaching that value. And once again in this case, this is gonna be equal to g of pi. We don't have any interesting discontinuities there or anything like that. So there's two big takeaways here. You can construct many different functions that would have the same limit at a point, and for a given function, you can take the limit at many different points, in fact an infinite number of different points. And it's important to point that out, no pun intended, because oftentimes we get used to seeing limits only at points where something strange seems to be happening.