Is my landlord required to test for asbestos before repairing ceiling?

[Sumpter Smartt]

I live in a rental that was built in '82 (or at least that's as far back as the deeds go on the county appraisal website). Anyway, the upstairs bathtub has started leaking into the laundry room below, and there's a small hole in the ceiling. The house has acoustic ceilings and since I'm buying a house now, I am familiar with the fact that there may be asbestos in the popcorn ceiling. Even though the manufacture of asbestos was banned in like '78, they continued to use it. The landlord is telling me the house isn't old enough for asbestos and I doubt they will test for it because their "handyman" says they don't need to. What options do I have? What if I test it myself and it comes back positive? I have a young child and I'm terrified of her being exposed to something like that.