Question #158022

If [x] is the greatest integer not greater than x, then determine the limit of [x ]when x goes to (1/2) is it exists


1
Expert's answer
2021-02-02T05:23:52-0500

Here, we have f(x)=[x]f(x)=[x] (greatest integer function)


The graph of this function looks like this:-




Here, we want to find limx12[x]\lim\limits_{x\to\frac{1}{2}} [x]


Here we calculate LHL and RHL separately.


LHL:-


limx12[x]=0\lim\limits_{x\to\frac{1}{2}^-} [x]=0 (As here x approaches 12\frac{1}{2} from left and so, [x] is 0)


RHL:-


limx12+[x]=0\lim\limits_{x\to\frac{1}{2}^+} [x]=0 (As here x approaches 12\frac{1}{2} from right and so, [x] is 0)


\therefore As LHL=RHL=0 , we know the limit exists and is:-



limx12[x]=0\color{purple} \lim\limits_{x\to\frac{1}{2}} [x]=0


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