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Limits of combined functions: piecewise functions

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

Video transcript

we are asked to find these three different limits I encourage you like always pause this video and try to do it yourself before we do it together so when you do this first one you might just try to find the limit as X approaches negative two of f of X and then the limit as X approaches negative two of G of X and then add those two limits together but you will quickly find a problem because when you find the limit as X approaches negative two of f of X it looks as we are approaching negative two from the left it looks like we're approaching one as we approach x equals negative two from the right it looks like we're approaching three so it looks like the limit as X approaches negative 2 of f of X doesn't exist and the same thing is true of G of X if we approach from the left it looks like we're approaching three which approach from the right it looks like we're approaching one but turns out that this limit can still exist as long as the limit as X approaches negative 2 from the left of the somme f of X plus G of X exists and is equal to the limit as X approaches negative 2 from the right of the sum f of X plus G of X so what are these things well as we approach negative 2 from the left f of X is approaching looks like 1 and G of X is approaching 3 so it looks like we're approaching 1 & 3 so it looks like this is approaching the sum is going to approach 4 and if we're coming from the right f of X looks like it's approaching 3 and G of X looks like it is approaching 1 and so once again this is equal to 4 and since the left and right handed limits are approaching the same thing we would say that this limit exists and it is equal to 4 now let's do this next example as X approaches 1 well we'll do the exact same exercise and once again if you look at the individual limits for f of X from the left and the right as we approach 1 this limit doesn't exist but the limit as X approaches 1 of the sum might exist so let's try that out so the limit as X approaches 1 from the left hand side of f of X plus G of X what is that going to be equal to as we approach so f of X as we approach 1 from the left it looks like this is going approaching - I'm just doing this for shorthand and G of X as we approach one from the left it looks like it is approaching zero so this will be approaching two plus zero which is two and then the limit as X approaches 1 from the right hand side of f of X plus G of X is going to be equal to well for f of X as we're approaching one from the right hand side looks like it's approaching a negative one and for G of X as we're approaching one from the right hand side looks like we're approaching a zero again and so here it looks like we're approaching negative one so the left and right hand limits aren't approaching the same value so this one does not exist and then last but not least X approaches 1 of f of X times G of X so we'll do the same drill limit as X approaches 1 from the left hand side of f of X times G of X well here and we could even use the values here we see we was approaching 1 from the left we are approaching 2 so this is 2 and when we're approaching 1 from the left here we're approaching 0 and so this is going to be 2 times we're going to be approaching 2 times 0 which is 0 and then we were approached from the right X approaches 1 from the right of f of X times G of X well we already saw when we're approaching 1 from the right of f of X we are approaching negative 1 but G of X approaching 1 from the right is still approaching 0 so this is going to be 0 again so this limit exists we get the same limit when we approach from the left and the right it is equal to 0 so these are pretty interesting examples because sometimes when you think that the component limits don't exist that that means that the sum or the product might not exist but this shows at least two examples where that is not the case
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