Limits of combined functions: piecewise functions | AP Calculus AB | Khan Academy

We are asked to find these three different limits. I encourage you, like always, to 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 2 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. Because 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 2 from the right, it looks like we're approaching 3. So it looks like the limit as x approaches negative 2 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 3; if we approach from the right, it looks like we're approaching 1. But it turns out that this limit can still exist as long as the limit as x approaches negative 2 from the left of the sum f of x plus g of x exists and is equal to the limit as x approaches negative 2 from the right of the sum f of x plus g of x.

So what are these things? Well, as we approach negative 2 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 3, and g of x looks like it is approaching 1. And so once again, this is equal to 4.

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? As we approach, so f of x as we approach 1 from the left, it looks like this is approaching 2. I'm just doing this for shorthand, and g of x as we approach 1 from the left, it looks like it is approaching 0. So this will be approaching 2 plus 0, which is 2.

And then the limit as x approaches 1 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 1 from the right-hand side, it looks like it's approaching negative 1, and for g of x as we're approaching 1 from the right-hand side, it looks like we're approaching 0 again. And so here it looks like we're approaching negative 1.

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, we have x approaching one of f of x times g of x. So we'll do the same drill: limit as x approaches 1 from the left-hand side of f of x times g of x. Well, here—and we could even use the values here—we see we were approaching 1 from the left; we are approaching 2, so this is 2. And when we're approaching 1 from the left here, we're approaching 0. And so this is going to be 2 times—we're going to be approaching 2 times 0, which is 0.

And then we approach from the right: x approaches 1 from the right of f of x times g of x. Well, we already saw when we're approaching 1 from the right of f of x, we are approaching negative 1, but g of x approaching 1 from the right is still approaching 0. So this is going to be 0 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.