You have to assign the vertices in the vertex group of that bone to adjust which vertices to be deformed by the bone, not a limit scale.
Do this, select the armature and SHFT+Select the mesh making sure that the mesh is the active object. Then go to edit mode, and on the right properties panel go to the vertex groups tab, and select the vertex group assigned to the bone of the character's arm. It should have the name of the bone.
Make sure no vertices are selected before selecting any vertex groups by pressing A once. Then select the vertex group for the bone.
The vertices assigned to that bone should appear. If some are showing up that you do not want to deform from that bone, deselect them.
After this, press CTRL+I to select inverse, and then press "Remove". This will then remove the vertices selected from that vertex group, so that they don't deform with that bone.
So I found the issue to be that you have an unnecessary bone in between the arm and the shoulder. By deleting this bone, the arm deforms properly.
Now the arm deforms properly: