Limits from graphs | Limits and continuity | AP Calculus AB | Khan Academy

So we have the graph of y equals f of x right over here, and we want to figure out three different limits. And like always, pause this video and see if you can figure it out on your own before we do it together.

All right, now first, let's think about what's the limit of f of x as x approaches 6. So as x, let me do this in a color you can see. As x approaches 6 from both sides, well, as we approach 6 from the left-hand side, from values less than 6, it looks like our f of x is approaching 1. And as we approach x equals 6 from the right-hand side, it looks like our f of x is once again approaching 1. In order for this limit to exist, we need to be approaching the same value from both the left and the right-hand side. And so here, at least graphically, so you never are sure with the graph, but this is a pretty good estimate. It looks like we are approaching one right over there.

Let me do the darker color now. Let's do this next one, the limit of f of x as x approaches four. So as we approach four from the left-hand side, what is going on? Well, as we approach 4 from the left-hand side, it looks like our function, the value of our function, it looks like it is approaching 3. Remember, you can have a limit exist at an x value where the function itself is not defined. The function, if you said what is f of 4, it's not defined, but it looks like when we approach it from the left, when we approach x equals 4 from the left, it looks like f is approaching 3. And when we approach 4 from the right, once again, it looks like our function is approaching 3. So here I would say, at least from what we can tell from the graph, it looks like the limit of f of x as x approaches 4 is 3, even though the function itself is not defined there.

Now let's think about the limit as x approaches 2. So this is interesting. The function is defined there, f of 2 is 2. Let's see when we approach from the left-hand side; it looks like our function is approaching the value of 2. But when we approach from the right-hand side, when we approach x equals 2 from the right-hand side, our function is getting closer and closer to 5. It's not quite getting to 5, but as we go from, you know, 2.1, 2.01, 2.001, it looks like our function, the value of our function, is getting closer and closer to five.

And since we are approaching two different values from the left-hand side and the right-hand side as x approaches two from the left-hand side and the right-hand side, we would say that this limit does not exist. So does not exist, which is interesting. In this first case, the function is defined at 6, and the limit is equal to the value of the function at x equals 6. Here, the function was not defined at x equals four, but the limit does exist. Here, the function is defined at f equal at x equals two, but the limit does not exist as we approach x equals two.

Let's do another function just to get more cases of looking at graphical limits. So here we have the graph of y equal to g of x, and once again, pause this video and have a go at it. See if you can figure out these limits graphically.

So first we have the limit as x approaches 5 of g of x. So as we approach 5 from the left-hand side, it looks like we are approaching this value. So let me see if I can draw a straight line that takes us. It looks like we're approaching this value, and as we approach 5 from the right-hand side, it also looks like we are approaching that same value. So this value, just eyeballing it off of here, looks like it's about 0.4. So I'll say this limit definitely exists; just when we're looking at a graph, it's not that precise. So I would say it's approximately 0.4. It might be 0.41; it might be 0.41456789; we don't know exactly just looking at this graph, but it looks like a value roughly around there.

Now, let's think about the limit of g of x as x approaches 7. So let's do the same exercise. What happens as we approach from the left, from values less than 7, 6.9, 6.99, 6.999? Well, it looks like the value of our function is approaching 2. It doesn't matter that the actual function is defined; g of 7 is 5, but as we approach from the left, as x goes 6.9, 6.99, and so on, it looks like our value of our function is approaching 2. And as we approach x equals 7 from the right-hand side, it seems like the same thing is happening. It seems like we are approaching 2.

And so I would say that this is going to be equal to 2. And so once again, the function is defined there, and the limit exists there, but the g of 7 is different than the value of the limit of g of x as x approaches 7.

Now let's do one more. What's the limit as x approaches 1? Well, we'll do the same thing from the left-hand side. It looks like we're going unbounded as x goes 0.9, 0.99, 0.999, 0.99999. It looks like we're just going unbounded towards infinity. And as we approach from the right-hand side, it looks like the same thing. The same thing is happening; we're going unbounded to infinity.

So formally, sometimes informally, people might say, oh, it's approaching infinity or something like that. But if we want to be formal about what a limit means in this context, because it is unbounded, we would say that it does not exist. Does not exist.