Question

If $f\left(x\right)=x(\sqrt{x}-\sqrt{x+1})$, then $f\left(x\right)$ is

• A. continuous but not differentiable at $x=0$
• B. differentiable at $x=0$
• C. not continuous at $x=0$
• D. None of the above

Answer 1

The correct option is C not continuous at $x=0$

$LHL={lim}_{x\to 0^{-}}f\left(x\right)={lim}_{x\to 0^{-}}x(\sqrt{x}-\sqrt{x+1})=$ does not exist

$RHL={lim}_{x\to 0^{+}}f\left(x\right)={lim}_{x\to 0^{+}}x(\sqrt{x}-\sqrt{x+1})=0$

Hence, is discontinuous at $x=0$ and hence, non-differentiable.