How to prove regular languages are closed under left quotient?
Assume $M$ is a deterministic finite state machine accepting $L$. Feed the word $w$ into $M$, which will land you in some state $q$. Construct a new machine $M&#39;$ which is the same as $M$ but has start state $q$. I claim that $M&#39;$ accepts $w^{-1}L$. 

Now prove it.