Example approximating limit graphically

The function H is defined for all real numbers, and they graph y is equal to H of x right over here; that's what they're showing us. They ask us what is a reasonable estimate for the limit as x approaches -7 of H of x, and they give us some choices for those reasonable estimates, including the possibility that the limit does not exist. So I encourage you to pause the video and see if you can work through this yourself.

All right, now let's think about what's going on. Something is interesting at x equal to -7. At x = -7, we have the function; it is defined, but it is not continuous with the rest of the function. H of -7 looks like a little bit; it looks like it's like -4. Something. So it's approximately -4. I don't know; it looks like negative 4.1 something, something, something. So that's what H of -7 is.

Sometimes there's this temptation to say, "Oh, whatever the function equals, that must be what the limit is." But this is actually a really good example for showing the difference that many times the limit exists and is approaching a value different than what the value that the function takes on at that point.

So let's think about the limit as x approaches -7 from smaller values of x. As x gets closer and closer to -7, it looks like the value of our function is approaching where we have this gap. It seems like our function is getting closer; the value of our function is approaching this value right over here. Similarly, if we were to approach from the left, so we’re approaching from more negative values, it looks like the value of our function is approaching that same thing.

Since it looks like it's approaching the same value whether we're coming from more negative values or less negative values, we know, or we can assume, or we have a good sense that this limit exists. We seem to be approaching the same value. Now, it's a value different than the actual value of the function there. The value of the function there is at -4.1 or something like that.

Now this looks like the function itself is approaching; this looks like, I don't know, 1, 1.2, 1.3, something like that. So now let's look at our choices. What's a reasonable estimate for this limit? Well, it's definitely not -7. The function is definitely not approaching y = -7. This is kind of a distractor choice because this is what x is approaching. X is approaching -7, but we want to know what is the function approaching. What is the value of H approaching as x approaches -7? So let's rule that one out.

-4.1 is an interesting choice because that's a good approximation for what the actual value of the function is there. The function is defined there, although it kind of jumps and is defined right over here; it's discontinuous. So this is more of what H; this could be a reasonable estimate for H of -7, what the function actually is defined to be at x = -7. But as we approach x equals -7, it does not seem to be approaching -4.1, so I'm going to rule that one out.

Now, 1.3—that is a pretty reasonable estimate for what the function seems to be approaching as x gets closer and closer to -7. You know, from negative 6.9, negative 6.99 from the right or from the left, negative 7.1, negative 7.01, -7.1, so on and so forth. It looks like the value of that function is approaching approximately 1.3, so I like that choice right over there.

1.8, well, that's just another choice, but this looks approximately where this is; it definitely looks closer to 1 than it does to 2, so I'd rule that one out. I would say that this limit does exist because it looks like we're approaching the same value whether we're coming from below or from above, so to speak.