In California, is there any limit as to how much a landlord can raise the rent and how much advance notice must the tenant be given?

Do you have a contract with the tenant? The amount of time you need to tell them about the cost of the raised rent should be in there. But regardless if you are going to raise the rent you should let them know about 2-3 mo. In advance so if they do not like the cost they can find a new place to live. I don't know what the rules are in California but I do know that in any place the amount of rent should be consistent with other rentals in the same status.

In California a landlord may raise the rent as much as he or she likes there are no rules set upon it. However if the increase is 10% then they must inform you 30 days before the increase takes place if it is greater then 10% then they must provide you with notice 60 days before it is to take affect.

There is a link to CA laws in a post at the top of this forum. You can't raise the rent except when a lease expires. Whether there is a limit might depend on the municipality. As for notice, it probably is specified in the lease,