0.999... = 1 (?)

Going off this one a bit:

If we take out the infinities, here's what happens:

lim (lim (E (1/x) from 0 to y) as y approaches b) as x approaches c

Then by this theorum:
E (1/x) from 0 to y = y/x

You get:
lim (lim (y/x) as y approaches b) as x approaches c
= lim (b/x) as x approaches c
= b/c

So then we end up with what we had before:
lim (b/c) as b and c approach infinity

As you said, differentiate.

lim db/dc as b and c approach infinity
= lim 1/1 as b and c approach infinity
= 1

*Still confused.*
Maybe a visual example like a graph may help.

If we take out the infinities (I'll use multiplication, as it is the same thing)


Nevermind what I said about differentiation though. It's not going to apply in this scenario anyway, as we have two limits. You have to evaluate one of the limits first, then the other. Period. Therefore, as we have infinity-limits here, we just have to evaluate those.

(I'm kind-of doing something illegal though: Infinity isn't really a number, so differentiating it does not really make sense. For this equation, it does not really matter, as we already found out that this equals to 0 from the other limit.)

Noah