I'm looking for an example of a concave function $g \colon [0,1] \to \mathbb{R}$, $g(0)=0$ such that:

$$\liminf_{x\to 0^+}\frac{g(x)}{-x\ln x}\neq \limsup_{x\to 0^+}\frac{g(x)}{-x\ln x}.$$

Moreover, is it possible that the upper limit is infinite while the lower limit is finite?