Those limes look like Persian, Tahiti, or Bearrs limes (all the same thing). It's the most common lime you find in markets.
Here in California, even this far north, most citrus grows great. It's not hot enough here on the coast in the summer to grow most grapefruit and some blood oranges, but I've got oranges of five varieties: Midknight, Trovita, Washington, Sanguinelli, Moro; eureka and meyer lemons; Bearrs, Kaffir (thai), and mexican limes; Minneoloa tangelo, Meiwa kumquat. The std Bearrs lime tree has grown quite large and is very prolific. I have to harvest it before the fruit is fully ripe or the weight will break major branches. More limes than I could possibly use, but I do prefer them to lemons. I am planning on replacing it with a true dwarf version.