Is there a limit to the number of tiles a city can work? I've noticed that I can purchase and use tiles several spaces away from my city. How far can I go?
Up to 3 hexes away, 36 hexes total
Plus one more for the hex the city is on.
All of the previous games in the Civilization Series used squares for the grid, and the maximum city radius was 2ish squares away. Take a look at the image below for what I mean by 2ish. (In words, the exact city radius is a 3x3 grid centered on the city, with one square from each corner removed.) That was a total of 20 squares, plus one more for the city square.
Civilization 4, and earlier
Your cities can get to hexes that are up to 3 away from them, although they all have to be contiguous, so you can't get to one that is 3 away until you have one that is two away connected to it. There is no limit to the number of tiles you can work (other than the limits on your population).
The maximum limit is five tiles, but thats for very large, isolated(more resorces) cities and usually your capital.