If the equity loan was written off the books in 2013, can the lender still sue for the equity loan amount?

It is not clear what you mean by the HELOC (Home Equity Line of Credit) being "written off the books." If a senior lienholder (usually the purchase money lender) forecloses and the auction sale price does not exceed the amount of the senior lien, then the junior lien (in this case the lien of the HELOC lender) is wiped out. A lien is created by a deed of trust (mortgage) that gives the lender the right to foreclose. However, this only means that the HELOC lender no longer has foreclosure of the property as a remedy to collect the loan. It may not wipe out the loan obligation itself. It depends on whether the junior loan was a "recourse" or "nonrecourse" loan. A recourse loan means that if foreclosure does not result in the loan being fully paid off, the lender can still sue in court for a money judgment in the amount of the remaining loan balance. California has what's called an antideficiency statute, which in effect makes all purchase money loans nonrecourse by restricting the lender to the remedy of foreclosure and barring a lawsuit to collect any remaining loan balance (deficiency). But the statute doesn't apply to second loans such as your HELOC. The HELOC promissory note and deed of trust probably specify that the loan is a recourse loan. But you should carefully read the documents, or have an attorney or real estate agent read them for you, to determine this for certain. If it is a recourse loan, then yes the lender can sue you for the remaining amount owed. If by, "written off the books" you mean that the lender has declared the loan balance as a loss, then it probably won't come after you. But legally, you still owe the money, so the bank could still sue you up to the date the statute of limitations expires, which is four years from the date the lender became legally entitled to sue. When a lender writes off something like a defaulted second mortgage, it's actually just showing that loan as a loss on its books. Written off loans haven't actually been forgiven or canceled, though, meaning you still owe the debt on them. Lenders writing off second mortgages or other loans still retain the right to collect against them. In many cases, written off loan debt is also sold to collection agencies that then try to collect against it.

You can have personal liability on a second if it is not used as purchase money or to improve the property.

Primary loan, no. Equity loan yes. Statute of limitations would expire this year.