Citrus are my favorite container plants.

If you want a lime, try Bearss (the regular, grocery store Persian type) as opposed to Key/Mexican, which is probably too frost sensitive for your area. I have read that all limes are very frost sensitive, but I have 2 Bearss--one in ground and one in a container-- and they are surprisingly tough and vigorous. Though I am in sunset zone 17, they have been through light frosts without extra protection. Being near your pool will help with microclimate management and humidity. If you want to keep your citrus in containers, get a dwarf or semi-dwarf. Same for Clementine, which should be really good with your hot Sac summers.

Regarding potting media, I would not use bagged soil or garden soil-- all too heavy and poorly draining. I have tried a few things, and the only really important thing is to use something that drains really well, or else you'll get root rot. Beware of overwatering, and water deeply to flush soil when you do water--to keep soil from building up crud. But do keep soil lightly moist (which is my problem--I'm stingy with water). Everyone has their own favorite container media, and I have used many different things, but probably the least labor-intensive is cactus mix with extra perlite in case you're lazy like me ;) but I am sure someone else will chime in.

Fertilize with nitrogen-rich formulas that contain iron, but do so judiciously.

Lastly--you will have to repot ( not pot up necessarily) at some point, but it may not be for a few years.

Also consider a lemon (lisbon is my fav--another tough one).

Check out the citrus forum--they know a lot over there.
HTH.

I agree with rickjames on all accounts. Bearss will be the best bet for a lime, and clementines will do well. However, Limes are definitely the most frost sensitive of the citrus. We had a big shrub in Walnut Creek that did great for many years and was unfortunately done in by the "all time" record temps of 1990, but I wouldn't necessarily stop that from using the -- usually, they were fine, although heavy frosts nipped just the youngest, greenest tips some years.

Still, in pots, lime trees aren't as beautiful as citrus with colorful fruit, such as tangerines, lemons, and such. Plus, limes are cheap and plentiful in the stores. There are some other wonderful container citrus that would look more Tuscan and be more colorful, and my suggestions would be: kumquat, Persian lime, and meyer lemon. But if you have your heard set on regular limes, go for it!

I love to see Red Valencias, Cara-Cara, Tarocco, Vainiglia Sanguigno, Moro and other pigmented flesh orange. Their fruit skins are the deepest colors and oftentimes splotched with red.

If you don't eat the fruits, just for decor or marmalades, then you can plant chinotto sour orange, the fruits last long, the canopy is dense and compact, very beautiful form and lovely contrasts with the fruits.