Asymptotics of a convolution

Question

For $r>1$ define the functions $$f(x)=|x|^{-1/2}\chi_{[-1,1]\setminus\{0\}}\quad\text{and}\quad g(x)=|x|^{-1/2r}(-\chi_{[-1,0)}+\chi_{(0,1]}).$$ I am interested in the asymptotic behavior of $h(x)=(f*g)(x)=\int f(y)g(x-y)~\mathrm{d}y$ as $x\searrow 0$. Numerical analysis suggests that $h(x)$ behaves like $x^{1-1/2-1/2r}$. Is that correct? If yes I would be interested in the value of $$\lim_{x\searrow0} x^{-1+1/2+1/2r}h(x).$$ It seems pretty much impossible to compute the convolution explicitly, at least Mathematica fails to do so.