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.

# Connecting limits and graphical behavior

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

## Video transcript

so we have the graph of y is equal to G of X right over here and I want to think about what is the limit as X approaches 5 of G of X well we've done this multiple times let's think about what G of X approaches as X approaches 5 from the left G of X is approaching negative 6 as X approaches 5 from the right G of X looks like it's approaching negative 6 so a reasonable estimate based on looking at this graph is that as X approaches 5 G of X is approaching negative 6 and it's worth noting that that's not what G of 5 is G of 5 is a different value but the whole point of this video is to appreciate all that a limit does a limit only describes the behavior of a function as it approaches a point it doesn't tell us exactly what's happening at that point what G of 5 is and doesn't tell us much about the rest of the function about the rest of the graph for example I could construct many different functions for which the limit as X approaches 5 is equal to negative 6 and they would look very different from G of X for example I could say the limit of f of X as X approaches 5 is equal to negative 6 and I can construct an f of X that does this that looks very different than G of X and in fact if you're up for it pause this video and see if you could do the same if you have some graph paper or even just sketch it well the key thing is that the behavior of the function as X approaches 5 from both sides from the left on the right it has to be approaching negative 6 so for example a function that looks like this so let me draw f of X and f of X that looks like this and is even defined right over there and then does something like this that would work as we approach from the left we're approaching negative 6 as we approach from the right we are approaching negative 6 you could have a function like this let's say the limit let's call it H of X as X approaches 5 is equal to negative 6 you could have a function like this maybe too defined up to there then it's a circle there and then it keeps going maybe maybe it's not defined at all for any of these values and then maybe down here it is defined for all X values greater than or equal to 4 and it just goes right through negative 6 so notice all of these all of these functions is X approaches 5 they all have the limit defined and it's equal to negative 6 but these functions all look very very very different now another thing to appreciate is for given function and let me delete these oftentimes we're asked to find the limit as X approaches some type of an interesting value so for example X approaches 5 5 is interesting right over here because we have this point discontinuity but you could take the limit on an infinite number of points for this function right over here you could say the limit of G of X as X approaches naught X equals as X approaches 1 what would that be positive ideon try to figure it out let's see as X approaches 1 from the left hand side it looks like we are approaching this value here and as X approaches 1 from the right hand side it looks like we are approaching that value there so that would be equal to G of 1 that is equal to G of 1 based on that would be a reasonable s that that's a reasonable conclusion to make looking at this graph and if we were estimate that G of 1 is looks like it's approximately negative five point one or five point two negative five point one we could find the limit of G of X as X approaches PI so pi is right around there as X approaches PI from the left we're approaching that value which just looks actually pretty close to what we just thought about and as we approach from the right we're approaching that value and once again in this case this is going to be equal to G of Pi we don't have any interesting discontinuities there or anything like that so there's two big takeaways here you can construct many different functions that would have the same limit at a point and for a given function you can take the limit at many different points in fact an infinite number of different points oh and it's important to point that to point that out no pun intended because oftentimes we get used to seeing limits only at points where something strange seems to be happening
AP® is a registered trademark of the College Board, which has not reviewed this resource.