I'm not using a calculator.

So,
lim x->0 of 0/x = 0
lim x->0+ of x/0 = infinity
lim x->0- of x/0 = -infinity
lim x->0 of x/x = 1
lim x->0 of (x/2)/x = 1/2
lim x->0 of nx/x = n (n!=0)

I know how to show 0 = 0, but how about the rest of these? How is infinity = 0? If n != 0, how is it 0, then? 0^0 or 0/0 cannot simply be described with one number.

An additional note:
lim x-> 0 of 0^x = 0
lim x-> 0 of x^0 = 1

All "modern algebra" does is lead me to believe that 0/0 is some set that includes {0,1} at least.

My brain hurts now. >_<

You are using limits to describe finite concepts. We already know 0 can be divided into 0 parts. Limit definitions are solid when finite definitions do not exist, eg diving by zero. Generally speaking, you apply the limit to the modifying operation, eg division, raising to a power, etc.

A limit can be applied anywhere. A limit only exists when the limit is finite, anyway. This statement really confuses me.


But you keep saying 0/0 = 0. 0 is neither -infinity nor infinity, so it is finite.


Each of the limits above are valid for evaluating 0/0. The "modifying operation" is rather ambiguous in this case. 0/0 could be split up to 0*1/0, or any other number of ways and we'd have a few operations to choose from.
Depending on how we approach 0/0, we can get infinitely different answers. Consider the equation (3x^2+nx)/x. At x=0, we have 0/0, and there is a removable discontinuity, but we can still evaluate the limit and it gives us n.

The function happens to be continuous. If we say that 0/0 = 0, we've broken the continuity.