Answer to Question #158022 in Real Analysis for Cypress

Question

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

Answer 1

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

The graph of this function looks like this:-

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

Here we calculate LHL and RHL separately.

LHL:-

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

RHL:-

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

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

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