We investigate the behaviour of the Stiltjes transformation $S_rf$ of an $f\in S'_+$ which has the $S$-asymptotic behaviour at $\infty$. As well we show that the domain of the convergence of $(S_rf)(z)$, $|z|\to\infty$, which is, by appropriate assumptions, of the form $\{re^{i\phi};\ r>0,\ |\phi|<\pi-\varepsilon\}$, $0<\varepsilon<\pi/2$, could not be enlarged to contain a half line $x+iy$, $y\neq0$, $x\in(-\infty,0)$.