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# Differentiability at a point: algebraic (function is 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 less differentiable at x equals three and they've defined it piecewise and we have some choices continuous not differentiable differentiable not continuous both continuous and differentiable neither continuous nor differentiable now one of these we can knock out right from the get-go it you cannot in order to be differentiable you need to be continuous there so you cannot have differentiable but not continuous so let's just rule that one out and now let's think about continuity so let's first think about continuity and frankly if it isn't continuous then it's not going to be differentiable so let's think about it a little bit so in order to be continuous f of these in a darker color F of three needs to be equal to the limit of f of X as X approaches three now what is F of three well see we fall into this category here because X is equal to three so 6 times 3 is 18 minus 9 is 9 so this is 9 so the limit of f of X as X approaches 3 needs to be equal to 9 so let's first think about the limit as we approach from the left hand side the limit as X approaches 3 X approaches 3 from the left hand side of f of X well when X is less than 3 we've fallen into this case so f of X is just going to be equal to x squared and so this is defined and continuous for all real numbers so we could just substitute the 3 in there so this is going to be equal to 9 now what's the limit of as we approach 3 from the right hand side of f of X well as we approach from the right this one right over here is f of X is equal to 6x minus 9 so if we just write 6x minus 9 and once again 6x minus 9 is defined and continuous for all real numbers so we could just pop a 3 in there and you get 18 minus 9 well this is also equal to 9 so the right and left hand and the left and right hand limits both equal nine which is equal to the value of the function there so it is definitely continuous so we can rule out we can rule out this choice right over there and now let's think about differentiability so in order to be differentiable so differentiable I'll just differentiable in order to be differentiable the limit as X approaches 3 of f of X minus F of 3 over X minus 3 needs to exist so see if we can evaluate this so first of all we know what F of 3 is F of 3 we've already evaluated this this is going to be 9 and let's see if we can evaluate this limit or let's see what this the limit is as we approach from the left hand side on the right hand side and if they approaching the same thing then we know that this slit then that same thing that they're approaching is the limit so let's first think about the limit as X approaches 3 from the left hand side so it's over X minus 3 and we have f of X minus 9 but as we approach from the left hand side this is f of X as X is less than 3 f of X is equal to x squared so this would be instead of f of X minus 9 I'll write x squared minus 9 and x squared minus 9 this is a difference of squares so this is X plus 3 times X minus 3 X plus 3 times X minus 3 and so these would cancel out we can say that this equivalent to X plus 3 as long as X does not equal 3 that's okay because we're approaching from the left and as we approach from the left well X plus 3 is is defined for all real numbers as continuous for all real numbers so we can just substitute the 3 in there so we could we would get a 6 so now let's try to evaluate the limit as we approach from the right hand side so once again it's f of X but as we approach from the right hand side f of X is 6x minus 9 that's our f of X and then we have minus f of 3 which is 9 so it's 6x minus 18 6x minus 18 well that's the same thing as 6 times X minus 3 and as we approach from the right well that's just going to be equal to that's just going to be equal to 6 so it looks like our our derivative exists there and it is equal the limit as X approaches 3 of all of this business is equal to 6 because the limit as we approach from the left and the right is also equal to 6 so this looks like we are both continuous and both continuous and differentiable