We used to call dead trees, broken down limbs etc. "widow makers", for a very good reason. Be careful.
I could have taken them down myself, but I called the pros for these because of their size and proximity to the house.
I have a bunch more that are not as big and farther away. I'll probably take a few down myself this fall, but most of them will be fine taking their natural course.
I hate to cut down a tree because even during all the stages of its death and decomposition it provides habitat for many different kinds of critters.