Question

Find :
$\underset{x\to 0}{lim}\frac{\left[x\right]}{x}$

Answer 1

We have,

${lim}_{x\to 0}\left(\frac{\left[x\right]}{x}\right)$

On taking R.H.L, we get

$R.H.L={lim}_{x\to 0^{+}}\left(\frac{\left[x\right]}{x}\right)$

$R.H.L=\left(\frac{0}{0^{+}}\right)$

$R.H.L=0$

On taking L.H.L, we get

$L.H.L={lim}_{x\to 0^{-}}\left(\frac{\left[x\right]}{x}\right)$

$L.H.L=\left(\frac{-1}{0^{-}}\right)$

$L.H.L=\infty$

So, $R.H.L\ne L.H.L$

Hence, limit does not exists.