  We study the weak limits of solutions to SDEs
\[dX_n(t)=a_n\bigl(X_n(t)\bigr)\,dt+dW(t),\] where the sequence $\{a_n\}$
converges in some sense to $(c_- 1\mkern-4.5mu\mathrm{l}_{x<0}+c_+
1\mkern-4.5mu\mathrm{l}_{x>0})/x+\gamma\delta_0$. Here $\delta_0$ is the Dirac
delta function concentrated at zero. A limit of $\{X_n\}$ may be a Bessel
process, a skew Bessel process, or a mixture of Bessel processes.
