My question in response to this answer: what would the finite automata look like for $L_1$ and $L_0$ in the answer?
I get how the languages are formed; however, since $M_L$ cannot remember how many times it has looped, how does $q$ branch off (if it does) into the two different DFAs for $L_0$ and $L_1$?
Definitions:
$L$ = infinite regular language
$q$ = state within the DFA for $L$, $M_L$, where $M_L$ loops
$L_1$ = {w in A | $q$ is visited an odd number of times}
$L_0$ = {w in A | $q$ is visited an even number of times}