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# Introduction to infinite limits

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.D (LO)
,
LIM‑2.D.1 (EK)
,
LIM‑2.D.2 (EK)

## Video transcript

in a previous video we've looked at these graphs this is y is equal to one over x squared this is y is equal to one over X and we explored what's the limit as X approaches zero in either of those scenarios and in this left scenario we saw as X becomes less and less negative as it approaches zero from the left hand side are the value of 1 over x squared is unbounded in the positive direction and the same thing happens as we approach X from the right as we come less and less positive but we are still positive the value of 1 over x squared becomes unbounded in the positive direction so in that video we just said hey one could say that this limit is unbounded but what we're going to do in this video is introduce new notation instead of just saying it's unbounded we could say hey from both the left and the right it looks like we're going to positive infinity so we can introduce this notation of saying hey this is going to infinity which you will sometimes see use some people would call this unbounded some people say it does not exist because it's not approaching some finite value while some people will use this notation of the limit going to infinity but what about this scenario can we use our new notation here well when we approach 0 from the left it looks like we're unbounded in the negative direction and when we approach 0 from the right we're unbounded in the positive direction so here you still could not say that the limit is approaching infinity because from the right it's approaching infinity but from the left it's approaching negative infinity so you would still say that this does not exist you could do one-sided limits here which if you're not familiar with I encourage you to review it on Khan Academy if you said the limit of 1 over X as X approaches 0 from the left hand side from values less than 0 well then you would look at this right over here and say well look it looks like we're going unbounded in the negative direction so you'd say this is equal to negative infinity and of course if you said the limit as X approaches 0 from the right of 1 over X well here your unbounded in the positive direction so that's going to be equal to positive infinity let's do an example problem from Khan Academy based on this idea in this notation so here it says consider graphs a B and C the dashed lines represent asymptotes which of the graphs agree with this statement that the limit as X approaches 1 of H of X is equal to infinity pause this video and see if you can figure it out alright let's go through each of these so we want to think about what happens at x equals 1 so that's right over here on graph a so as we approach x equals 1 so let me write this so the limit let me do this for the different graphs so for graph a the limit as X approaches 1 from the left that looks like it's unbounded in the positive direction that equals infinity and the limit as X approaches 1 from the right well that looks like it's going to negative infinity that equals negative infinity and since these are going in two different directions you wouldn't be able to say that the limit is X approaches 1 from both directions is equal to infinity so I would rule this one out now let's look at choice B what's the limit as X approaches 1 from the left and of course these are off of H of X I've got to write that down so of H of X right over here well as we approach from the left we are going to looks like we're going to positive infinity and it looks like the limit of H of X as we approach 1 from the right is also going to positive infinity and so since we're approaching you to say the same direction of infinity you could say this for B so B meets the constraints but let's just check C to make sure well you can see very clearly x equals 1 then as we approach it from the left we go to negative infinity and as we approach from the right we go to positive infinity so this once again would not be approaching the same infinity so you would rule this one out as well
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