I suppose these are the equations with infinity that are universally considered correct:
- ∞ = ∞
- ∞ + n = ∞
- ∞ * n = ∞
- n/∞ = 0
Where n can be any possible value.
These equations can be rearranged to give the following results:
- ∞ - ∞ = 0
- ∞ - ∞ = n
- ∞ / ∞ = n
- ∞ * 0 = n
Where n can be any possible value.
But can n also be infinite?
If so the following final derivations can be made(in no particular order):
- ∞ + ∞ = ∞
- ∞ - ∞ = n (where -∞ <= n <= ∞)
- ∞ * ∞ = ∞
- ∞ / ∞ = n (where -∞ <= n <= ∞)
Are these statements valid? Also the interesting thing here is that in the final equations, 1. and 3. both have a single value of infinity whereas 2. and 4. Can have any possible value, including -∞, 0 and ∞. Does that mean 1. and 3. are not undefined?
Let us follow the convention that an expression with $\infty$ is "defined" (in the extended reals) if: when you replace each $\infty$ with any function/sequence whose limit is $\infty$, and each real number with any function/sequence with that limit, the limit of the entire expression is always the same real number or divergence to $\infty$ or $-\infty$.
Then you are right that $\infty+\infty$ and $\infty*\infty$ are defined and equal $\infty$. And you are right that $\infty-\infty$ and $\infty/\infty$ (and $0*\infty$) are undefined. (If you need proofs for these and can't find them elsewhere, let me know.)
Because of this, it is wrong to write something like "$\infty-\infty=n$ where $n$ is any number", because $\infty-\infty$ is undefined. It's not equal to anything. And we don't usually use the equals sign in a way that would allow multiple values at once.
I will interpret this as "are there functions where $\infty-\infty$, $\infty/\infty$ and $\infty*0$ correspond to limits that diverge to infinity?" Yes. For examples: ${\displaystyle \lim_{n\to\infty}}(2n)-(n)=\infty$, ${\displaystyle \lim_{n\to\infty}}(n^2)/(n)=\infty$, and ${\displaystyle \lim_{n\to\infty}}(n^2)*(1/n)=\infty$.