Decidability of equivalence problem with limit
You can still reduce $L_0$ to $EQ_{\lim}$. Given a Turing machine $m$, construct a Turing machine $m'$ which deletes its input and then transfers control to $m$, and let $n'$ denote a (fixed) Turing machine which halts on no input. Then $m \in L_0$ iff $\langle m',n' \rangle \in EQ_{\lim}$.