 Welcome back. Today we are going to be looking at limits and we're going to be specifying limits from the left and limits from the right. And I think the hardest part about doing limits this way is just the notation. So once you get the notation down, these usually aren't too bad to look at. So limit from the left. You want to think about a number line. So what numbers are the furthest to the left? And those are numbers that are negative. So that's why on here for the notation we have a little minus sign up there. So that's how I always remember that that's from the left, because the negative numbers are on the left of the number line. Then here when we're looking at from the right, again thinking about the numbers that are on the right side of a number line, those numbers are all positive. So that's why here there's a little plus sign on the limit. Okay, so now that we have the notation down, let's take a look at a couple of graphs. So I have a graph here. And of course the interesting part is at zero, because we have something going on here that's suspicious. So this graph here is looking nice and curvy, and then it goes like a linear graph. So given the following graph of f of x determine each limit, the first one I want to look at then is x is approaching zero with the little minus sign. Pause the video. Is that from the left or from the right? It's from the left as you look up here. So I'm going to want to be looking on the left side of zero, which is over here in the negative direction. So as I am kind of close to zero but coming into it, what is my y value approaching as my x's are approaching zero? Those are approaching zero. Okay, so let's look on the right side. Let me change colors. So the plus again means from the right. If we follow and come into zero from the right side, bam, right there, what does it look like my y values are approaching now? And I know it's kind of hard to tell by the grid lines here, but it looks like that's one. Okay, now because these two limits are not the same, we could say that the limit as x approaches zero without a plus or a minus of our function does not exist. Only because these two values are different. Okay, let's take a look at a formula now, and I'm going to stop the piece right here. So we've got g of x is equal to the piece by its function x minus four for x less than or equal to negative one and x squared plus five for negative one is less than or equal to x less than or equal to one. Sorry, strictly less than. Anyway, look at what I said. Look at what I wrote, not what I said. So we're including negative one in the top function. We're not including negative one in the bottom function. There we go. It's easy, right? So looking at this function, what is an interesting value that the limit that we should look at for our limit? Well, obviously negative one because that's where our function is breaking at. Okay, so part A of course asks about the limit as x approaches negative one. And again, I see a plus sign. That plus sign means from the right. So thinking about then negative one, what kind of numbers are to the right of negative one? Well, numbers that are bigger than negative one. So obviously it makes sense then to use the second function for this one. So I'm just going to make a note here. Second. Okay, so if I look at the values then as we're approaching negative one. I know negative one is not included in this function, but if we go ahead and input negative one into our function, then that's going to give us a value of 6. Now the reason why that's legit to do is because this is a limit. This is not a function value. So we know that as we keep coming closer and closer and closer to negative one, we're going to have a hole here, right? Because it's not included. So if you think about a graph that it's not going to be actually equal to there, but it's going to be coming really, really, really, really close to that value, and that's exactly what a limit is. Okay, now as we're approaching from the right, or sorry, we just did from the right, from the left, and you notice here there's a little minus sign. What kind of numbers are to the left of negative one? Well, these are numbers that are smaller than negative one. So obviously for this one then, I'm going to be using the first function. So for this one, negative one is included. So I'm sure it's a little bit more intuitive that it's okay to plug negative one into this particular function. And in this case, the limit value and the function value are the same. But even if that were not included, if this had just been strictly less than negative one, then we still could put negative one into the function because it's going to come really, really, really darn close to it. It's just never going to actually equal it. So the limit value at, as we approach negative one from the left, is going to give us negative five. So again, I know I didn't ask for it, but we could then say that the limit as x approaches negative one, no left or right, just overall, of our function g of x, is again, does not exist. So make it rid of that equal sign. Does not exist, because those two values are not the same. All right, thank you for watching.