If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Estimating limit values from graphs

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.C (LO)
,
LIM‑1.C.2 (EK)
,
LIM‑1.C.3 (EK)
,
LIM‑1.C.4 (EK)

## Video transcript

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 just 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 and 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 1 right over there I'm doing darker color now let's do this next one the limit of f of X as X approaches 4 so as we approach 4 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 exists 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 two but when we approach from the right hand side when we approach x equals two from the right hand side our function is getting closer and closer to five it's not quite getting to five but as we go from you know 2.1 2.0 1 to 0.0001 it looks like our function the value of our function is getting closer and closer to 5 and since we are approaching two different values from the left hand side and the right hand side as X approaches 2 from the left hand side on 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 4 but the limit does exist here the function is defined at f equal at x equals 2 but the limit does not exist as we approach x equals 2 let's do another function just to get more cases of looking at graphical limits so here we have the graph of y is 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 so 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 and so this value just eyeballing it off of here it looks like it's about point 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 point 4 1 it might be 0 point 4 1 4 5 6 7 8 9 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 is we approach from the left from values less than seven six point nine six point nine nine six point nine nine nine well it looks like the value of our function is approaching two 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 six point nine six point nine nine and so on it looks like our value of our function is approaching two and as we approach x equals seven from the right hand side it seems like the same thing is happening it seems like we are approaching two and so I would say that this is going to be equal to two and so once again the function is defined there and the limit exists there but the G of seven 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 one well we'll do the same thing from the left hand side it looks like we're going unbounded as X goes point nine zero point nine nine zero point nine nine nine zero point nine nine nine nine nine 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 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