How to patch a tin roof built in gutter?

Question

gutter looks like a trough at the edge of the roof

Answer 1

Last time I patched anything on a tin roof, we used a piece of tin and some roofing tar.

If you wanted to get fancy, you could solder a piece of tin over the hole.

Answer 2

If the gutter isn't rusted through everywhere, then it's probably leaking at the seams. Clean the seam thouroughly removing any old caulking. Do not apply tar. Tar will go on too thick which will cause leaves and stuff to pile up and clog up the gutter quicker, plus, it will just dry out and crack and leak again and make it harder to fix the next time.

Home Depot carries a product called SeamerMate by Amerimax (or at least they used to) which is semi self-leveling and can be applied over gutter seams or holes. It's better than any other kind of caulking or even soldering and lasts many years.

If there is a lot of rust it may be time to replace the gutters.