There is a right way to saw off a limb from a tree.
To begin with, make an undercut near the tree trunk.
If you don't do this, when the limb falls, it will tear off the bark on the underside of the remaining branch as well as a strip from the trunk itself.
The second cut can be made at a convenient place on the branch but close to the trunk, leaving a stub.
Then cut the stub off flush with the tree.
As the final step.
cover the exposed raw wood with a good grade of paint.