You can have any pair $-\infty\le a\le b\le+\infty$ as limit inferior and limit superior, choosing a suitable concave function $g$ (and whatever is the nonnegative concave function $h(x)$ in place of $-x\log x$, satisfying $h(0)=0$ and $h'(0)=+\infty$). Idea: construct $g$ piecewise linear. To do so define inductively a strictly decreasing sequence $x_k\to0$, and a decreasing sequence of values $y_k=g(x_k)\to0$, thus defining intervals $J_k:=[x_{k+1},x_k]$ where $g$ is affine. The only constraint to get a concave function is that the slope of $g$ on each interval $J_k$ has to be smaller than $y_k/x_k$, and increasing wrto $k$; this is not an obstruction to approach any limit inferior and limit superior for $g(x)/h(x)$, provided that $x_{k+1} > 0$ is choosen small enough.

You can have any pair $-\infty\le a\le b\le+\infty$ as limit inferior and limit superior, choosing a suitable concave function $g$ (and whatever is the nonnegative concave function $h(x)$ in place of $-x\log x$, satisfying $h(0)=0$ and $h'(0)=+\infty$). Idea: construct $g$ piecewise linear. To do so define inductively a strictly decreasing sequence $x_k\to0$, and a decreasing sequence of values $y_k=g(x_k)\to0$, thus defining intervals $J_k:=[x_{k+1},x_k]$ where $g$ is affine. The only constraint to get a concave function is that the slope of $g$ on each interval $J_k$ has to be smaller than $y_k/x_k$, and increasing wrto $k$; this is not an obstruction to approach any limit inferior and limit superior for $g(x)/h(x)$, provided that $x_{k+1} > 0$ is chosen small enough.

You can have any pair $-\infty\le a\le b\le+\infty$ as limit inferior and limit superior, choosing a suitable concave function $g$ (and whatever is the nonnegative concave function $h(x)$ in place of $-x\log x$, satisfying $h(0)=0$ and $h'(0)=+\infty$). Sketch of the idea: construct $g$ piecewise linear. To do so define inductively a strictly decreasing sequence $x_k\to0$, and a decreasing sequence of values $y_k=g(x_k)\to0$, thus defining intervals $J_k:=[x_{k+1},x_k]$ where $g$ is affine. The only constraint to get a concave function is that the slope of $g$ on each interval $J_k$ has to be smaller than $y_k/x_k$, and decreasing wrto $k$; this is not an obstruction to approach any limit inferior and limit superior for $g(x)/h(x)$, provided that $x_{k+1} > 0$ is choosen small enough.

You can have any pair $-\infty\le a\le b\le+\infty$ as limit inferior and limit superior, choosing a suitable concave function $g$ (and whatever is the nonnegative concave function $h(x)$ in place of $-x\log x$, satisfying $h(0)=0$ and $h'(0)=+\infty$). Idea: construct $g$ piecewise linear. To do so define inductively a strictly decreasing sequence $x_k\to0$, and a decreasing sequence of values $y_k=g(x_k)\to0$, thus defining intervals $J_k:=[x_{k+1},x_k]$ where $g$ is affine. The only constraint to get a concave function is that the slope of $g$ on each interval $J_k$ has to be smaller than $y_k/x_k$, and increasing wrto $k$; this is not an obstruction to approach any limit inferior and limit superior for $g(x)/h(x)$, provided that $x_{k+1} > 0$ is choosen small enough.