Proving that the continuation of a non-regular language is not ω-regular
If $L'$ is regular then it is easy to extend a DFA for $L'$ to a deterministic B&uuml;chi automaton for $L$. For the other direction, start with a (general) B&uuml;chi automaton for $L$. Call a state winning if the automaton accepts starting at the state upon reading $x^\omega$. Remove all $x$ transitions and make winning states accepting to obtain an NFA for $L'$.