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# Limits of piecewise functions

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

## Video transcript

let's think a little bit about limits of piecewise functions that are defined algebraically like our f of X right over here pause this video and see if you can figure out what these various limits would be some of them are one-sided and some of them are regular limits or two-sided limits alright let's start with this first one the limit as X approaches 4 from values larger than equaling 4 so that's that what that Plus tells us and so when X is greater than 4 our f of X is equal to square root of x so as we are approaching 4 from the right we are really thinking about this part of the function and so this is going to be equal to the square root of 4 even though right at 4 our f of X is equal to this we are approaching from values greater than 4 we're approaching from the right so we would use this part of our function definition and so this is going to be equal to 2 now what about our limit of f of X as we approach 4 from the left well then we would use this part of our function definition and so this is going to be equal to 4 plus 2 over 4 minus 1 which is equal to 6 over 3 which is equal to 2 and so if we want to say what is the limit of f of X as X approaches 4 well this is a good scenario here because from both the left and the right as we approach x equals 4 we are approaching the same value and we know that in order for the 2 sided limit to have a limit you have to be approaching the same thing from the right and the left and we are and so this is going to be equal to 2 now what's the limit as X approaches 2 of f of X well as X approaches 2 we are going to be completely in this scenario right over here now interesting things do happen at x equals 1 here our denominator goes to 0 but at x equals 2 this part of the curve is going to be continuous so we can just substitute the value it's going to be 2 plus 2 over 2 minus 1 which is 4 over 1 which is equal to 4 let's do another example so we have another piecewise function and so let's pause our video and figure out these things all right now let's do this together so what's the limit as X approaches negative 1 from the right so if we're approaching from the right when we are greater than or equal to negative 1 we are in this part of our piecewise function and so we would say this is going to approach this is going to be 2 to the negative 1 power which is equal to 1/2 what about if we're approaching from the left well if we're approaching from the left we're in this scenario right over here where to the left of x equals negative 1 and so this is going to be equal to the sign because we're in this case for our piecewise function of negative 1 plus 1 which is a sine of 0 which is equal to 0 now what's the two-sided limit as X approaches negative 1 of G of X well we're approaching two different values as we approach from the right and as we approach from the left and if our one-sided limits aren't approaching the same value well then this limit does not exist does not exist and what's the limit of G of X as X approaches 0 from the right well if we're talking about approaching 0 from the right we are going to be in this case right over here 0 is definitely in this interval and over this interval this right over here is going to be continuous and so we can just substitute x equals 0 there so it's going to be 2 to the 0 which is indeed equal to 1 and we're done
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