Is there a limit to the number of tiles a city can use in Civ 5?

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.

In square or isometric (1-4) Civ games, you can work 2 tiles away except the 4 corners.

In hexagon (5-6) Civ games, you can work 3 tiles away.

You can access strategic or luxury goods within your border, but cannot work tiles outside of work limit. You can share tiles between cities, and transfer them by clicking on a tile in the city screen. You can also use cultural borders of another city to work a tile further away. With Civ 5-6, you could work a tile 3 away within having any tiles 2 away because it is within the cultural border of another city.

Culture Levels (numbers are for Normal Speed): Quick is 1/2x, Epic is 3/2x, Marathon is 3x.

0. None (0)
1. Poor (0)
2. Fledgling (10)
3. Developed (100)
4. Refined (500), Civ4 Tutorial (1,000)
5. Influential (5,000), Civ4 Tutorial (10,000)
6. Legendary (50,000), Civ4 Tutorial (100,000)

Cultural Levels of the Civ4 Tutorial is different from the actual game in Normal speed. You could mod the XML so that Poor may require cultural points so that new cities start without any workable tiles; this is not recommended, as without Cultural Borders, anyone could capture by walking into a new city without any military units to defend.