One-sided limits from tables

- [Instructor] The function F is defined over the real numbers. This table gives select values of F. We have our table here. For these X values, it gives the corresponding F of X. What is a reasonable estimate for the limit of F of X as X approaches one from the left? So pause this video and see if you can figure it out on your own. Alright, now let's work through this together. So 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, and 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, this is saying, let me put a little arrow here, this is the limit of F of X as X approaches one from the left, from the left. So from the left, how do we know that? Well that's what that little negative tells us. It tells us we're approaching one from values less than one. If we were approaching one 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 one from the left. 0.9, which is already pretty close to one, then we get even closer to one from the left. Notice, these are all less than one, but they're getting closer and closer to one. And so what we really wanna look at is what does F of X approach as X is getting closer and closer to one, from the left, from the left. And a key realization here is, if we're thinking about general limits, not just from one direction, then we might wanna 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 of X at X equal one confuse us. Sometimes and oftentimes, the limit is approaching a different value than the value of the function at that point. So let's look at this. At 0.9, F of X is 2.5. When we get even closer to one from the left, we go to 2.1. When we get even closer to one from the left, we're getting even closer to two. So a reasonable estimate for the limit as X approaches one from the left of F of X, it looks like F of X right over here is approaching two. 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 was approaching a whole number, then two could be a reasonable estimate right over here. Although it doesn't have to be two, 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. So 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 two from the left? So this is confusing. You see these two negative signs. This first negative sign tells us we're approaching negative two. We wanna say, what happens we're approaching negative two, and we're gonna approach once again from the left. So lucky for us, they have values of X that are approaching negative two from the left, so this is X approaches negative two from the left, so that is happening right over here. So that's these values. So notice, this is negative 2.05, then we get even closer, negative 2.01, then we get even closer, negative 2.002, and these are from the left because these are values less than negative two, but they're getting closer and closer to negative two. And so let's see, when we're a little bit further, F of 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 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 two, without getting there, it looks like this is getting unbounded. It looks like it's becoming infinitely negative. And so technically, it looks like this is, I would write this is unbounded, and so if this was a multiple choice question, technically you would say the limit as X approaches negative two from the left does not exist, does not exist. If someone asked the other question, if they said, what is the limit as X approaches negative two from the right of F of X, well then you would say, alright, well here are values approaching negative two from the right, so this is X approaching negative two 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 we wanna think about what is the value of the function approaching, as your X is approaching that value, as X is approaching, in this case, negative two, from the right. So as we're getting closer and closer to negative two from values larger than negative two, it looks like F of X is getting closer and closer to negative four, which is F of negative two, 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. And in general, if you are approaching different values from the left than from the right, then you would say, a that point, the limit of your function does not exist, and we have seen that in other videos.