If you could include your thought process in determining why it's regular it would help me a lot.
$L_1 = (0^*(10)^*11)$
$L_2 = \{ \langle M \rangle \mid M \text{ is a Turing machine that halts on all inputs from }L_1 \}$
$L_3 = \{ x \in \{0,1\}^* \mid \exists y \in L_2. xy \in L_1 \}$
Why is $L_3$ regular? It's a set of strings, I need to determine if there's a DFA that can accept it. Do I even care about $L_2$ and $L_1$ in this case?