There is a man at the door.
A man is at the door.
I was asked what's the difference and when do you use which?
I have 2 methods of problem solving the first is run as many examples as possible in my head and ty to figure out a pattern. The second would be to go to a book but in the case of this example finding this answer may prove to be difficult.
I generally prefer the first method of running over examples.
(Since I was forced to answer and my first answer "Stop splitting hairs." was deamed unacceptable, I answered that the first sentence sounds like general information and that the person we're talking to should probably go answer the door. The sencond sentence sounds more like you're asking for an explanation or I'm assuming you know why there is a man at the door.)
So, how do you usually answer tough grammar questions?