If I recently quit my job and my health insurance premium is up at the end of the month, are they supossed to let me know in writing that my insurance is lapsed?

Since you have recenly quit your job and your health care insurance premium is due to be renewed at the end of this month, your former employer at the time of the presumed exit interview was required to provide you with all information for you to continue with your health plan under "COBRA". I would safeguard your interests so that your insurance does not lapse.