One-sided limits from tables | Limits and continuity | AP Calculus AB | Khan Academy

The function ( f ) is defined over the real numbers. This table gives select values of ( f ). We have our table here; for any of these ( x ) values, it gives the corresponding ( f(x) ). What is a reasonable estimate for the limit of ( f(x) ) as ( x ) approaches one from the left?

So pause this video and see if you can figure it out on your own.

All right, now let's work through this together. The important, the first thing that is really important to realize is when you see this ( x ) approaches one and you see this little negative superscript here, this does not mean approaching negative one. So this does not mean negative one. Sometimes your brain just sees a one and that little negative sign there; you're like, "Oh, this must be a weird way of writing negative one," or you don't even think about it. But it's not saying that; it's saying—let me put a little arrow here—this is the limit of ( f(x) ) as ( x ) approaches 1 from the left.

From the left! How do we know that? Well, that's what that little negative tells us. It tells us we're approaching 1 from values less than one. If we're approaching 1 from the right—from values greater than one—that would be a positive sign right over there.

So let's think about it. We want the limit as ( x ) approaches one from the left, and lucky for us, on this table we have some values of ( x ) approaching 1 from the left: 0.9, which is already pretty close to 1. Then we get even closer to 1 from the left; notice these are all less than 1, but they're getting closer and closer to 1.

And so what we really want to look at is the val—what does ( f(x) ) approach as ( x ) is getting closer and closer? Let me write it as ( x ) is getting closer and closer to 1 from the left. A key realization here is if we're looking—if we're thinking about general limits—not just from one direction, then we might want to look at from the left and from the right. But they're asking us only from the left, so we should only be looking at these values right over here.

In fact, we shouldn't even let the value of ( f(x) ) at ( x ) equal one confuse us. Sometimes, and often times, the limit is approaching a different value than the actual value of the function at that point.

So let's look at this: at 0.9, ( f(x) ) is 2.5. When we get even closer to 1 from the left, we go to 2.1. When we get even closer to 1 from the left, we're getting even closer to 2.

So a reasonable estimate for the limit as ( x ) approaches 1 from the left of ( f(x) ) looks like ( f(x) ) right over here is approaching 2. We don't know for sure; that's why they're saying, "What is a reasonable estimate?" It might be approaching 2.01, or it might be approaching 1.999.

On Khan Academy, these will often be multiple choice questions, so you have to pick the most reasonable one. It would not be fair if they gave a 1.999 as a choice and 2.01; but if you were saying, "Hey, maybe this is approaching a whole number," then 2 could be a reasonable estimate right over here.

Although it doesn't have to be 2; it could be 2.01258. It might be what it is actually approaching.

So let's try another example here. Here it does look like there's a reasonable estimate for the limit as we approach this value from the left.

Now it says the function ( f ) is defined over the real numbers. This table gives select values of ( f ). Similar to the last question, what is a reasonable estimate for the limit as ( x ) approaches negative 2 from the left?

So this is confusing. You see these two negative signs. This first negative sign tells us we're approaching negative 2. We want to say what happens when we're approaching negative 2, and we're going to approach once again from the left.

So lucky for us, they have values of ( x ) that are approaching negative 2 from the left. This is ( x ) approaches negative 2 from the left; so that is happening right over here.

Notice this is negative 2.05, then we can even closer negative 2.01, then we get even closer: negative 2.002. These are from the left because these are values less than negative 2, but they're getting closer and closer to negative 2.

So let's see, when we're a little bit further, ( f(x) ) is negative 20. We get a little bit closer; it's negative 100. We get even a little bit closer; it goes to negative 500.

So it looks—it would be reasonable, and we don't know for sure; this is just giving us a few sample points for this function. But if we follow this trend as we get closer and closer to negative 2 without getting there, it looks like this is getting unbounded. It looks like it's becoming infinitely negative.

So technically, it looks like this is—I would write this as unbounded. If this was a multiple choice question, technically you would say this: the limit as ( x ) approaches negative 2 from the left does not exist.

It does not exist! If someone asked the other question, if they said, "What is the limit as ( x ) approaches negative 2 from the right of ( f(x) )?" well then you would say, "All right, well here are values approaching negative 2 from the right."

So this is ( x ) approaching negative 2 from the right, right over here. And remember, when you're looking at a limit, sometimes it might be distracting to look at the actual value of the function at that point.

So you want to think about what is the value of the function approaching as ( x ) is approaching that value. As ( x ) is approaching—in this case, negative 2—from the right, as we're getting closer and closer to negative 2 from values larger than negative 2, it looks like ( f(x) ) is getting closer and closer to negative 4, which is ( f ) of negative 2.

But that actually seems like a reasonable estimate. Once again, we don't know absolutely for sure just by sampling some points, but this would be a reasonable estimate.

In general, if you are approaching different values from the left than from the right, then you would say at that point the limit of your function does not exist. And we've seen that in other videos.