Determining limits of composite functions: internal limit doesn't exist

0:00- [Instructor] All right, let's get a little bit more practice taking limits of composite functions. Here, we want to figure out what is the limit as x approaches negative one of g of h of x? The function g, we see it defined graphically here on the left, and the function h, we see it defined graphically here on the right.

0:18Pause this video and have a go at this. All right, now your first temptation might be to say, all right, what is the limit as x approaches negative one of h of x, and if that limit exists, then input that into g. If you take the limit as x approaches negative one of h of x, you see that you have a different limit as you approach from the right than when you approach from the left.

0:41So your temptation might be to give up at this point, but what we'll do in this video is to realize that this composite limit actually exists even though the limit as x approaches negative one of h of x does not exist. How do we figure this out? Well, what we could do is take right-handed and left-handed limits.

1:02Let's first figure out what is the limit as x approaches negative one from the right hand side of g of h of x? Well, to think about that, what is the limit of h as x approaches negative one from the right hand side? As we approach negative one from the right hand side, it looks like h is approaching negative two.

1:32Another way to think about it is this is going to be equal to the limit as h of x approaches negative two, and what direction is it approaching negative two from? Well, it's approaching negative two from values larger than negative two. H of x is decreasing down to negative two as x approaches negative one from the right.

1:55So it's approaching from values larger than negative two of g of h of x. G of h of x. I'm color coding it to be able to keep track of things. This is analogous to saying what is the limit, if you think about it as x approaches negative two from the positive direction of g?

2:18Here, h is just the input into g. So the input into g is approaching negative two from above, from the right I should say, from values larger than negative two, and we can see that g is approaching three. So this right over here is going to be equal to three. Now, let's take the limit as x approaches negative one from the left of g of h of x.

2:51What we could do is first think about what is h approaching as x approaches negative one from the left? As x approaches negative one from the left, it looks like h is approaching negative three. We could say this is the limit as h of x is approaching negative three, and it is approaching negative three from values greater than negative three.

3:15H of x is approaching negative three from above, or we could say from values greater than negative three, and then of g of h of x. Another way to think about it, what is the limit as the input to g approaches negative three from the right? As we approach negative three from the right, g is right here at three, so this is going to be equal to three again.

3:51So notice the right hand limit and the left hand limit in this case are both equal to three. So when the right hand and the left hand limit is equal to the same thing, we know that the limit is equal to that thing. This is a pretty cool example, because the limit of, you could say the internal function right over here of h of x, did not exist, but the limit of the composite function still exists.