I used to live in just such a place. Went to the city center to apply for a library card, thinking "of course I can get a card here, I live in Foo City and this is the Foo City Public Library." I was asked for my street address, and she pulled out a binder of street names to check (yes it was analog, in the year 2016 A.D.). I was not within the city limits and was denied a card.