Limits of composite functions: external limit doesn't exist | AP Calculus | Khan Academy

So, over here I have two functions that have been visually or graphically defined. On the left here, I have the graph of g of x, and on the right here, I have the graph of h of x. What I want to do is figure out what is the limit of g of h of x as x approaches one. Pause this video and see if you can figure that out.

Alright, now let's do this together. Now, the first thing that you might try to say is, "Alright, let's just figure out first the limit as x approaches 1 of h of x." When you look at that, what is that going to be? Well, as we approach 1 from the left, it looks like h of x is approaching 2, and as we approach from the right, it looks like h of x is approaching 2. So, it looks like this is just going to be 2.

Then we say, "Okay, well maybe we could then just input that into g." So, what is g of 2? Well, g of 2 is 0, but the limit doesn't seem defined. It looks like when we approach 2 from the right, we're approaching 0, and when we approach 2 from the left, we're approaching negative 2. So, maybe this limit doesn't exist, but if you're thinking that, we haven't fully thought through it.

Because what we could do is think about this limit in terms of both left-handed and right-handed limits. So, let's think of it this way. First, let's think about what is the limit as x approaches 1 from the left-hand side of g of h of x.

Alright, when you think about it this way, if we're approaching 1 from the left, right over here we see that we are approaching 2 from the left, I guess you could say, or we're approaching 2 from below. The thing that we are inputting into g of x is approaching 2 from below. So, if you approach 2 from below, right over here, what is g approaching? It looks like g is approaching negative 2. So this looks like it is going to be equal to negative 2, at least this left-handed limit.

Now let's do a right-handed limit. What is the limit as x approaches 1 from the right hand of g of h of x? Well, we can do the same exercise. As we approach one from the right, it looks like h is approaching 2 from below, from values less than 2. So, if we are approaching 2 from below—because remember, whatever h is outputting is the input into g—if the thing that we're inputting into g is approaching 2 from below, that means that g, once again, is going to be approaching negative 2.

So this is a really, really, really interesting case where the limit of g as x approaches 2 does not exist. But because on h of x, when we approach from both the left and the right-hand side, h is approaching 2 from below, we just have to think about the left-handed limit as we approach 2 from below, or from the left, on g.

Because in both situations we are approaching negative 2. And so that is going to be our limit. When the left-handed and the right-hand limit are the same, that is going to be your limit. It is equal to negative 2.