Limits of combined functions: piecewise functions

- [Instructor] We are asked to find these three different limits. I encourage you like always, pause this video and try to do it yourself before we do it together. So when you do this first one, you might just try to find the limit as x approaches negative two of f of x and then the limit as x approaches negative two of g of x and then add those two limits together. But you will quickly find a problem, 'cause when you find the limit as x approaches negative two of f of x, it looks as we are approaching negative two from the left, it looks like we're approaching one. As we approach x equals negative two from the right, it looks like we're approaching three. So it looks like the limit as x approaches negative two of f of x doesn't exist, and the same thing's true of g of x. If we approach from the left, it looks like we're approaching three. If we approach from the right, it looks like we're approaching one. But it turns out that this limit can still exist as long as the limit as x approaches negative two from the left of the sum, f of x plus g of x, exists and is equal to the limit as x approaches negative two from the right of the sum, f of x plus g of x. So what are these things? Well, as we approach negative two from the left, f of x is approaching, looks like one, and g of x is approaching three. So it looks like we're approaching one and three. So it looks like this is approaching. The sum is going to approach four. And if we're coming from the right, f of x looks like it's approaching three and g of x looks like it is approaching one. Once again, this is equal to four. And since the left and right handed limits are approaching the same thing, we would say that this limit exists and it is equal to four. Now let's do this next example as x approaches one. Well, we'll do the exact same exercise. And once again, if you look at the individual limits for f of x from the left and the right as we approach one, this limit doesn't exist. But the limit as x approaches one of the sum might exist, so let's try that out. So the limit as x approaches one from the left hand side of f of x plus g of x, what is that going to be equal to? So f of x, as we approach one from the left, looks like this is approaching two. I'm just doing this for shorthand. And g of x, as we approach one from the left, it looks like it is approaching zero. So this will be approaching two plus zero, which is two. And then the limit, as x approaches one from the right hand side of f of x plus g of x is going to be equal to. Well, for f of x as we're approaching one from the right hand side, looks like it's approaching negative one. And for g of x as we're approaching one from the the right hand side, looks like we're approaching zero again. Here it looks like we're approaching negative one. So the left and right hand limits aren't approaching the same value, so this one does not exist. And then last but not least, x approaches one of f of x times g of x. So we'll do the same drill. Limit as x approaches one from the left hand side of f of x times g of x. Well, here, and we can even use the values here. We see it was approaching one from the left. We are approaching two, so this is two. And when we're approaching one from the left here, we're approaching zero. We're gonna be approaching two times zero, which is zero. And then we approach from the right. X approaches one from the right of f of x times g of x. Well, we already saw when we're approaching one from the right of f of x, we're approaching negative one. But g of x, approaching one from the right, is still approaching zero, so this is going to be zero again, so this limit exists. We get the same limit when we approach from the left and the right. It is equal to zero. So these are pretty interesting examples, because sometimes when you think that the component limits don't exist that that means that the sum or the product might not exist, but this shows at least two examples where that is not the case.