Limits of combined functions | Limits and continuity | AP Calculus AB | Khan Academy

So let's find the limit of f of x times h of x as x approaches 0.

All right, we have graphical depictions of the graphs y equals f of x and y equals h of x. We know from our limit properties that this is going to be the same thing as the limit as x approaches 0 of f of x times the limit as x approaches 0 of h of x.

Let's think about what each of these are. So let's first think about f of x right over here. As x approaches 0, notice the function itself isn't defined there. But we see when we approach from the left, the function seems to be approaching the value of negative one right over here. As we approach from the right, the function seems to be approaching the value of negative one.

So the limit here, this limit is negative one. As we approach from the left, we're approaching negative one. As we approach from the right, the value of the function seems to be approaching negative one.

Now, what about h of x? Well, h of x, we have down here. As x approaches zero, the function is defined at x equals zero. It looks like it is equal to one. The limit is also equal to one. We can see that as we approach it from the left, we are approaching one. As we approach from the right, we are approaching one.

As we approach x equals zero from the left, the function approaches one. As we approach x equals zero from the right, the function itself is approaching 1. It makes sense that the function is defined at x equals 0 and the limit as x approaches 0 is equal to the value of the function at that point because this is a continuous function.

So this is 1, and negative 1 times 1 is going to be equal to negative 1. So that is equal to negative 1.

Let's do one more. All right, so these look like continuous functions. We have the limit as x approaches zero of h of x over g of x. Once again, using our limit properties, this is going to be the same thing as the limit of h of x as x approaches 0 over the limit of g of x as x approaches 0.

Now, what's the limit of h of x as x approaches zero? Let's see. As we approach zero from the left, our function seems to be approaching four. As we approach x equals zero from the right, our function also seems to be approaching 4. That’s also what the value of the function is at x equals 0.

That makes sense because this is a continuous function. The limit as we approach x equals 0 should be the same as the value of the function at x equals 0. So this top is going to be 4.

Now, let's think about the limit of g of x as x approaches 0. From the left, it looks like the value of the function is approaching 0. As x approaches 0 from the right, the value of the function is also approaching 0. This also happens to be g of zero. g of zero is also zero.

This makes sense that the limit and the actual value of the function at that point is the same because it's continuous. So this also is zero.

But now we're in a strange situation. We have to take 4 and divide it by 0. So this limit will not exist because we can't take 4 and divide it by 0.

Even though the limit of h of x as x approaches 0 exists and the limit of g of x as x approaches zero exists, we can't divide four by zero. So this whole entire limit does not exist.

It does not exist, and actually, if you were to plot h of x over g of x, if you were to plot that graph, you would see it even clearer that that limit does not exist. You would actually be able to see it graphically.