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# Differentiability at a point: algebraic (function isn't differentiable)

AP.CALC:
FUN‑2 (EU)
,
FUN‑2.A (LO)
,
FUN‑2.A.1 (EK)
,
FUN‑2.A.2 (EK)

## Video transcript

is the function given below continuous / differentiable at x equals one and they define the function G piecewise right over here and then they give us a bunch of choices continuous but not differentiable differentiable but not continuous both continuous and differentiable neither continuous nor differentiable and like always pause this video and see if you could figure this out so let's do step by step so first let's think about continuity so for continuity for G to be continuous at x equals 1 that means that G of 1 that means that G of 1 must be equal to the limit as X approaches 1 of G of G of X well G of 1 what is that going to be G of 1 we're going to fall into this clock this case 1 minus 1 squared is going to be 0 so if we can show that the limit of G of X as X approaches 1 is the same as G of 1 is equal to 0 that we know we're continuous there let's do the left and right-handed limits here so if we do the left handed limit limit and that's especially useful because we're in these different clauses here and as we approach from the left and the right-hand side so as X approaches 1 from the left hand side of G of X well we're going to be falling into this situation here as we approach from the left as X is less than 1 so this is going to be the same thing as that that's what G of X is equal to when we are less than one as we're approaching from the left well this thing is is defined and it's continuous for all real numbers so we can just substitute 1 in for X and we get this is equal to 0 so so far so good let's do one more of these let's approach from the right-hand side as X approaches 1 from the right hand side of G of X well now we're falling into this case so G of X if we're to the right of 1 if we're great values greater than or equal to 1 it's going to be X minus 1 squared but once again X minus 1 squared that is defined for all real numbers as continuous for all real numbers so we could just pop that one in there you get 1 minus 1 squared well that's just 0 again so the left-hand limit the right-hand limit are both equals 0 which means that the limit is equal the limit is of G of X is X one is equal to zero which is the same thing as G of one so we are good with continuity so we can rule out all the ones that are saying that they are it's not continuous so we could rule out that one and we can rule out that one right over there so now let's think about whether it is differentiable so differentiability so differentiability alright differentiability ability did I let's see that's a long word differentiability all right differentiability what needs to be true here well we have to have a defined limit as X approaches one for f of X minus f of 1 over well let me be careful it's not f its G if it's G so we need to have a defined limit for G of X minus G of 1 over X minus 1 and so let's just try to evaluate this limit from the left and right hand sides and we could simplify it we already know that G of 1 is 0 so that's just going to be 0 so we just need to find the limit as X approaches 1 of G of x over X minus 1 or see if we can find the limit so let's first think about the limit as we approach from the left hand side of G of X over X minus G of X over X minus 1 well as we approach from the left hand side G of X is that right over there so we should we could write this instead of writing G of X we could write this as X minus 1 X minus 1 over X minus 1 and as long as we aren't equal to 1 this thing is going to be equal as long as X does not equal 1 X minus 1 over X minus 1 is just going to be 1 so this limit is going to be 1 so that was that one worked out now let's think about the limit as X approaches 1 from the right hand side of once again I could write G of X minus G of 1 but G of 1 is just 0 so I'll just write G of X over X minus 1 well what's G of X now well it's X minus 1 squared so instead of writing G of X I could write this as X minus 1 squared over X minus one and so as long as X does not equal one and we're just doing the limit we're saying as we approach one from the right hand side well this expression right over here you have X minus one squared divided by X minus one well that's just going to be give us X minus One X minus one squared divided by X minus one is just going to be X minus one and this limit well this expression right over here is going to be continuous and defined for for sure all X's that are not equaling one actually let me let me well it was it was before it was this X minus one squared over X minus one this thing right over here as I said it's not defined for x equals one but it is defined for anything not for x does not equal one and we're just approaching one and if we wanted to simplify this expression it would get we this would just be I think I just did this but I'm just making sure I'm doing it right this is going to be the same thing as that for X not being equal to one well this is just going to be zero you could just evaluate when X is equal to 1 here this is going to be equal to zero and so notice you get a different limit for this definition of the derivative as we approach from the left hand side or the right hand side and that makes sense this graph is going to look something like we have a slope of 1 so it's going to look something like this and then right when X is equal to 1 and the value of our function is 0 it looks something like this it looks something like this and so the graph is continuous the graph for sure is continuous but our slope coming into that point is 1 and our slope right when we leave that point is 0 so it is not differentiable over there so it is continuous but not continuous but not differentiable
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