Calculus: Manipulating Infinity: Nothing x Everything = Something

Have you ever wondered what happens when you divide Zero by Infinity? Or multiply Infinity by Zero? Well, using the concept of the Limit from Calculus we can manipulate 0 and ∞ to find out the answer to these, and other, very interesting questions.

Before we can begin, we must first agree on one thing. This won't be hard for anyone who has taken first year Calculus, but I'll explain it so the rest of you don't fall off the wagon at the very beginning. Here it is:

lim_{x → ∞}¹/_x = 0

This is read "the limit of one divided by X, as X approaches infinity, equals zero." If you who have taken calculus, this is obvious, and you can jump ahead. For the rest of you, think about ¹/_x. What happens to it as X gets larger and larger (ie: as it "approaches infinity")?

Well, ¹/₁₀ = 0.1.
¹/_1,000 = 0.001.
¹/_{1 billion} = 0.000000001.

As you can see, as X gets larger and larger, the value of ¹/_x gets closer and closer to zero. Thus, if X gets infinitely large, then ¹/_x would get infinitely close to zero. Thus, ¹/_x = 0 when X approaches infinity. Okay, you're all caught up!

Now that we have another way to represent 0 (namely, lim_{x → ∞}¹/_x), we can do all sorts of stuff. Let's start simple:
What is 0 · 0?
Well, 0 · 0 = lim_{x → ∞} (¹/_x) · (¹/_x)
Which simplifies to lim_{x → ∞}¹/_x²
Which = 0.
So, 0 · 0 = 0. No surprises there! But know that you understand the method, the other ones are easy to solve.

Here is another obvious one.
What is ∞ · ∞?
∞ · ∞ = lim_{x → ∞} x · x = lim_{x → ∞} x^{2 ₌} ∞
Thus, ∞ · ∞ = ∞.

Less obvious...
What is ⁰/_∞?
⁰/_∞ = lim_{x → ∞} (¹/_x) ÷ x = lim_{x → ∞}¹/_x² = 0
Thus, ⁰/_∞ = 0.

Inverse of the previous:
What is ^∞/₀?
^∞/₀ = lim_{x → ∞} x ÷ (¹/_x) = lim_{x → ∞} x² = ∞
Thus, ^∞/₀ = ∞.

Now for the strange:
What is ⁰/₀?
⁰/₀ = lim_{x → ∞} (¹/_x) ÷ (¹/_x) = lim_{x → ∞}^x/_x = 1
Thus, ⁰/₀ = 1.

And for the strangest of all:
What is ∞ · 0?
∞ · 0 = lim_{x → ∞} x · (¹/_x) = lim_{x → ∞}^x/_x = 1
Thus, ∞ · 0 = 1.
So, everything times nothing equals something?

Now that is quite a surprise! Note that this new equation, ∞ · 0 = 1, can be solved for ∞ in terms of 1 and 0 to yield:

∞ = ¹/₀

Curiouser and curiouser! Division by zero is supposed to be undefined. Apparently not in to world of the infinite!

And, according to this last form of the equation, everything equals something divided by nothing. I would be a bad mathematician to suggest that this implies that everything in existence is One and the same Something. So i wont. But will someone please do it, because it sounds really cool?