Question

The value of $\underset{x\to 0}{lim}\left(\right[x]+\sqrt{\{x}\})$ is (where $[.]$ and $\left\{\right\}$ denote the greatest integer function and fractional part function respectively )

• A. $2$
• B. $1$
• C. $0$
• D. limit does not exist

Answer 1

The correct option is C $0$

Given: $\underset{x\to 0}{lim}\left(\right[x]+\sqrt{\{x}\})$
$L.H.L.=\underset{x\to 0^{-}}{lim}\left(\right[x]+\sqrt{x-\left[x\right]})$
$=-1+\sqrt{0-(-1)}$
$=0$
$R.H.L.=\underset{x\to 0^{+}}{lim}\left(\right[x]+\sqrt{x-\left[x\right]})$
$=0+\sqrt{0-0}$
$=0$

$\therefore L.H.L=R.H.L.$