Question

Find the left hand and right hand limits of the greatest integer function $f\left(x\right)=\left[x\right]=$ greatest integer less than or equal to $x$, at $x=k$, where $k$ is an integer. Also, show that $\underset{x\to k}{lim}f\left(x\right)$ does not exist.

Answer 1

REF.Image.

Given $f\left(x\right)=\left[x\right]$

$\underset{x\to k}{lim}f\left(x\right)=$

Let us consider ${lim}_{x\to k^{+}}f\left(x\right)={lim}_{x\to k^{+}}\left[x\right]=k$

$\underset{x\to k^{-}}{lim}f\left(x\right)=\underset{x\to k^{-}}{lim}\left[x\right]-k-1$

$\therefore \underset{x\to k^{+}}{lim}f\left(x\right)\ne Ltf\left(x\right)$

$\therefore$ Limit doesnot exist