Connecting limits and graphical behavior | Limits and continuity | AP Calculus AB | Khan Academy

So, we have the graph of y is equal to g of x right over here, and I want to think about what is the limit as x approaches 5 of g of x. Well, we've done this multiple times. Let's think about what g of x approaches as x approaches 5.

From the left, g of x is approaching negative 6. As x approaches 5 from the right, g of x looks like it's approaching negative 6. So, a reasonable estimate, based on looking at this graph, is that as x approaches 5, g of x is approaching negative 6.

And it's worth noting that that's not what g of 5 is. g of 5 is a different value. But the whole point of this video is to appreciate all that a limit does. A limit only describes the behavior of a function as it approaches a point; it doesn't tell us exactly what's happening at that point, what g of 5 is, and it doesn't tell us much about the rest of the function, about the rest of the graph.

For example, I could construct many different functions for which the limit as x approaches 5 is equal to negative 6, and they would look very different from g of x. For example, I could say the limit of f of x as x approaches 5 is equal to negative 6, and I can construct an f of x that does this, that looks very different than g of x.

And in fact, if you're up for it, pause this video and see if you could do the same. If you have some graph paper or even just sketch it. Well, the key thing is that the behavior of the function as x approaches 5 from both sides, from the left and the right, it has to be approaching negative 6.

So, for example, a function that looks like this—let me draw f of x—a function that looks like this and is even defined right over there and then does something like this, that would work. As we approach from the left, we're approaching negative six; as we approach from the right, we are approaching negative six.

You could have a function like this—let's say the limit let's call it h of x as x approaches 5 is equal to negative 6. You could have a function like this; maybe it's defined up to there, then you have a circle there, and then it keeps going.

Maybe it's not defined at all for any of these values, and then, maybe down here, it is defined for all x values greater than or equal to 4, and it just goes right through negative 6. So, notice all of these functions as x approaches 5. They all have the limit defined, and it's equal to negative 6, but these functions all look very, very, very different.

Now, another thing to appreciate is for a given function—and let me delete these—oftentimes we're asked to find the limits as x approaches some type of an interesting value. So, for example, x approaches 5. 5 is interesting right over here because we have this point of discontinuity, but you could take the limit on an infinite number of points for this function right over here.

You could say the limit of g of x as x approaches—not x equals—as x approaches 1. What would that be? Positive? Try to figure it out. Let's see: as x approaches 1 from the left-hand side, it looks like we are approaching this value here, and as x approaches 1 from the right-hand side, it looks like we are approaching that value there.

So that would be equal to g of 1. That is equal to g of 1 based on that might be a reasonable conclusion to make, looking at this graph. If we were to estimate that g of 1 looks like it's approximately negative 5.1 or 5.2, negative 5.1.

We could find the limit of g of x as x approaches pi. So, pi is right around there. As x approaches pi from the left, we're approaching that value, which just looks actually pretty close to the one we just thought about, and as we approach from the right, we're approaching that value.

And once again, in this case, this is going to be equal to g of pi. We don't have any interesting discontinuities there or anything like that.

So, there are two big takeaways here. You can construct many different functions that would have the same limit at a point, and for a given function, you can take the limit at many different points. In fact, an infinite number of different points.

And it's important to point that out—no pun intended—because oftentimes we get used to seeing limits only at points where something strange seems to be happening.