One-sided limits from tables

the function f is defined over the real numbers this table gives select values of F we have our table here for any 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 1 from the left so pause this videos and see if you can figure it out on your own alright now let's work through this together so the important the first thing that is really important to realize is when you see this X approaches 1 and you see this little negative superscript here this does not mean approaching negative 1 so this does not mean negative 1 sometimes your brain just sees a 1 and that little negative sign there and you're like oh this must be a weird way of writing negative 1 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 1 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 1 from values less than 1 if we were approaching 1 from the right from values greater than 1 that would be a positive sign right over there so let's think about it we want the limit as X approaches 1 from the left and lucky for us on this table we have some values of X approaching 1 from the left is 0.9 that 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 Vallot what is f of 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 from the left and a key realization here is if we were 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 of X at X equal 1 confuse sometimes and oftentimes the limit is approaching a different value than the actual 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 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 of X it looks like X it looks like f of 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 point 0 1 or it might be approaching 1 point 9 9 9 on Khan Academy these will often be multiple-choice questions so you have to pick them the most reasonable one it would not be fair if they gave a 1 point 9 9 9 is a choice and 2 point 0 1 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 point 0 1 2 5 8 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 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 gonna approach once again from the left so lucky for us they have values of X that are approaching negative 2 from the left so this is X approaches negative 2 from the left and so that is happening right over here so that's these values so notice this is negative 2.05 then we can leave a closer negative 2.01 then we get even closer negative two point zero zero two 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 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 tune 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 and so technically it looks like this is I would write this as unbounded unbounded and so if this was a multiple choice question and technically you would say this the limit as X approaches negative 2 from the left does not exist and 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 of X well then you would say all right well here our value is approaching negative 2 from the right so this is X approaching negative 2 from the right right over here and remember you 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 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 2 from the right so as we're getting closer and closer to negative 2 from values larger than negative 2 it looks like f of 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 these would this 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 at that point the limit of your function does not exist and we've seen that in other videos