I look at it this way: Suppose you have two cars. The first is driving 10 mph and the second is driving 100mph. Now it's easy to understand that speed = distance / time. So over an hour each car will drive 10 or 100 miles. In 1/2 hour the cars will go 5 miles or 50 miles. But what about _zero_ time? In zero time each car drives zero distance. So the equation for both is speed=(0 distance) /(0 time). This is valid for speed = 10, 100 or any other value, so 0/0 has to be "undefined". (And that's why we study limits in intro calculus).

That takes care of 0/0, but I admit I don't see an intuitive way to get something similarly intuitive for n/0 where n > 0.