Death is inevitable, and an important mechanic designed to kill the egos of players. However, it can be fun to have your player characters die, albeit rarely.
As a DM, I've made it my mission to seek out the player characters and watch them die as quickly as possible, be it by falling, monsters, or maybe even by falling off a horse. The mission has some caveats though, in that the players have to agree that the death was fair, and not unforeseen (I cannot simply throw many monsters at them).
I feel like the above mission is a good assessment of one of the primary goals of the DM, but I'm not sure myself.
We also see times where PCs are on the ground, and helpless while the others are trying to actively fend off the other assailants, and as DM you cannot help but save the character.
What time is it appropriate to kill a PC?
Think about time in the session, time in the campaign (storyline), and time in the encounter (end of the cave of death, or near the beginning ).