When someone is clearly alive we tend to understand.
We also tend to understand when someone is clearly dead and ready to bury. There is no brain function, there is no movement in the body (including no breathing) and there is no flow of blood (the heart is not beating). Other signs of life are gone as well. the body is cold, dry, pale, either stiff with rigor mortis or flaccid.
Dying is a period of time between living and death. Some medical dictionaries define dying as the last stage of life. From a medical point of view dying is a process while death is an event. The process of dying is marked by a person having a disorder that is 1) untreatable, 2) progressive and that 3) ends in the event of death. When a person has elements 1 & 2 that are expected to end in 3, we say that the person is dying. One complicating factor is that, even in the year 2013, we can not always be certain. Many of us can think of examples where people were thought to be dying and ended up recovering and continuing in the land of the living. There are also examples of those who are thought to be be dying but the process takes much longer than anticipated. All people with incurable, progressive, chronic diseases, in a sense, are dying. But we do not usually consider incurable as dying, especially if there are still treatments that can slow down the progress of the condition. So, for the purpose of discussion we will limit dying to having an untreatable, progressive condition that is expected to result in the event of death within, at most, a few weeks. Even with this definition, we can still be wrong.