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# One-sided limits from graphs

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 if we were to ask ourselves what is the value of our function approaching as we approach X as we approach x equals 2 from values less than x equals 2 so as you imagine as we approach x equals 2 so x equals 1 x equals 1.5 x equals 1.9 x equals 1.999 x equals one point nine nine nine nine nine nine nine nine what is f of X approaching and we see that f of X seems to be approaching seems to be approaching this value seems to be approaching this value right over here it seems to be approaching five and so the way we would denote that is the limit the limit of f of X as X approaches two and we're going to specify the direction as X approaches 2 from the negative direction we put the negative as a superscript after the two to denote the direction that we're approaching this is not a negative two we're approaching 2 from the negative direction we're approaching 2 from values less than 2 we're getting closer and closer to two but from below one point nine one point nine nine one point nine nine nine nine nine as X gets closer and closer to from those values what is f of X approaching and we see here that it is approaching it is approaching five it is approaching five but what if we were asked a different question or I guess the the natural other question what is the limit what is the limit of f of X as X approaches 2 from values greater than two so this is a little superscript positive right over here so now we're going to approach x equals two but we're going to approach it from this direction x equals three x equals two point five x equals two point one x equals two point zero one x equals two point zero zero zero one we're going to get closer and closer to but we're coming from values that are larger than two so here when x equals three f of X is here when x equals two point five f of X f of X is here when x equals two point oh one f of X looks like it's right over here so in this situation we're getting closer and closer we're getting closer and closer to f of X equaling one it never does quite equal that actually then just has a jump discontinuity but it seems to be approaching this seems to be the limiting value when we approach X from values greater that when we approach 2 from values greater than 2 so this right over here is equal to 1 and so this is and so when we think about limits in general the only way that a limit at 2 will actually exist is if both of these limits these both of these one-sided limits are actually the same thing in this situation they aren't as we approach 2 from values below 2 we are proceeding 5 and is we approach 2 from values above 2 the function seems to be approaching 1 so in this case the limit let me write this down the limit of f of X as X approaches 2 from the negative direction does not equal does not equal the limit of f of X as X approaches 2 from the positive direction and since this is since this is the case that they're not equal the limit does not exist the limit as X approaches 2 in general of f of X so the limit of f of X is X the limit of f of X as X approaches 2 does not exist does not exist in order for to have existed these two things would have had to been equal to each other for example if someone were to say what is the limit what is the limit of f of X as X approaches 4 as X approaches 4 well then we can think about the the two one-sided limits the left the the one-sided limit from below and the one-sided limit from above so we could say well let's see the limit of f of X as X approaches 4 from below so let me draw that so what we care about x equals 4 as x equals 4 from below so when x equals 3 we're here where f of f of 3 is negative 2 f of 3.5 seems to be right over here f of 3 point 9 seems to be right over here F of 3 point 9 9 9 we're getting closer and closer to our function equaling equaling negative five so this the limit as we approach for from below this one-sided limit from the left we could say this is going to be equal to negative five and if we were to ask ourselves the limit of f of X as X approaches 4 from the right from values larger than four X approaches 4 from the right well same exercise f of five gets us here f of five f of 4.5 seems right around here f of 4.1 seems right about here f of four point zero one seems right around here and even F of four is actually defined but we're getting closer and closer to it and we see once again we are approaching five we are approaching five even if F of four was not defined on either side we would be approaching five or I sorry we would be approaching negative five so this is also approaching negative five and since the limit from the left-hand side is equal to the limit from the right hand side we can say so these two things are equal and because these two things are equal we know that the limit of f of X as X approaches four is equal to five let's look at a few more examples so let's ask ourselves let's ask ourselves the limit the limit of f of X now this is our new f of X depicted here as X approaches 8 and let's approach 8 from the left as f as X approaches 8 from values less than 8 so what's this going to be equal to it I encourage you to pause the video to try to figure it out yourself so X is getting closer and closer to 8 so if X is 7 f of 7 is here if X is 7.5 F of 7.5 is here so it looks like our value of f of X is getting closer and closer and closer to 2 3 so it looks like the limit of f of X as X approaches 8 from the negative side is equal to 3 what about from the positive side what about the limit of f of X as X approaches 8 from the positive direction or from the right side well here we see as X as X is 9 this is our f of X as X is 8 point 5 this is our F of 8 point 5 it seems like we're pro King it seems like we're approaching f of X equaling one so notice these two limits are different so the limit actually does the the non one-sided limit or the two-sided limit does not exist at f of X or at as we approach eight so let me write that down the limit of f of X as X approaches 8 because these two things are not the same value this does not exist does not does not exist let's do one more example in here they're actually asking us a question the function f is graphed below what appears to be the value of the one-sided limit the limit of f of X this is f of X as X approaches 2 from the negative direct or sorry it's X approach is negative 2 from the negative direction so this is the negative 2 from the negative direction so we care what happens as X approaches negative 2 we see f of X is actually undefined right over there but let's see what happens as we approach from the negative direction or as we approach from values less than negative 2 or as we approach from the left as we approach from the left F of negative 4 is right over here so this is F of negative 4 F of negative 3 is right over here negative 3 is right over there F of negative 2 point 5 seems to be right over here we seem to be getting closer and closer to f of X be equal to 4 at least visually so I would say that it looks at least graphically the limit of f of X as X approaches 2 from the negative Direction is equal to 4 now if we also asked ourselves the limit of f of X as X approaches negative 2 from the positive direction we would get a similar result where now we're going to approach from when x is 0 f of X seems to be right over here when X is 1 f of X is right over here when X is 1 point negative 1 point 9 9 when it exercises well this is when X is negative 1 f of X is there when X is negative 1 point 9 f of X seems to be good right over here so once again we seem to be getting closer and closer to 4 because the left handed limit and the right handed limit same value because both one-sided limits are approaching the same thing we can say that the limit of f of X as X approaches 2 I'm sorry as X approaches negative 2 and this is from both direction since from both directions we get the same limiting value we can say that the limit exists there and it is equal to 4