Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?

If the characteristic of $k$ is not $2$, $(t^2+\epsilon)=(t+\epsilon/2)^2$. So write $t'=t+\epsilon/2$. Them we are looking at the $t'^2$adic completion of $k[\epsilon][[t']]$. If the characteristic is $2$, then $(t^2+\epsilon)^2=t^4$. So this is just the $t$adic completion. Alternately, you can observe that a sequence is Cauchy with respect to this ideal if and only if it is Cauchy with respect to $t$, and the same for sequences converging to $0$, so the completions ae the same. 

