In this examplewe have exists,x= 1 is in the domainoff(x), but.
One-SidedLimitsand ClosedIntervals
Let’s recallour basicsquareroot function,.
Sincethe domainof isx≥ 0, we see that that doesnot exist.Specifically, we cannot
find openintervalsaroundx= 0 that satisfythe limit definition.Howeverwe do note that as we approachx
= 0 from the right-handside,we see the successivevaluestendingtowardsx= 0. This exampleprovides
somerationalefor how we can define one-sidedlimits.
Definition. We say that the right-hand limitof a functionf(x) ataisb, writtenas , if for
everyopenintervalNofb, thereexistsan openinterval containedin the domainof such
that is in for every in
For the exampleabove,we write
Similarly, we say that the left-handlimitof ataisb, writtenas , if for everyopen
intervalNofbthereexistsan openinterval containedin the domainof suchthat
is inNfor everyxin
Example1:
Find
The graphhas a discontinuityatx= 0 as indicated: