In mathematics, "$\infty - \infty$" is an indetermination because $\infty$ can be reached by lots of different ways, and we can define some of them as "bigger" than the others. Counter-intuitively, we also have that $\infty = \infty + \infty = \infty^2 = e^\infty = \infty^\infty$. What a mess !
Is there a way to "classify" the different infinities, in order to apply classical algebra to infinities ? Is there some kind of "infinity-algebra" ? For instance, can we define:
$$\infty_0 := \lim_{x\rightarrow +\infty} x$$
We could consider that "$x\rightarrow +\infty$" $ \Leftrightarrow $"$x\rightarrow \infty_0$", and thus:
$$\lim_{x\rightarrow \infty_0} x^2 = \infty_0^2$$
We would also have:
$$\lim_{x\rightarrow 0} \frac{1}{x} = \lim_{x\rightarrow \infty_0} x = \infty_0 =\ "\frac{1}{0}"$$
and so on.
Yes. Kind of. Such "limit" is called "germ". So what you propose is the germ of function $f(x)=x$ at infinity.
This kind of algebras is called Hardy fields. Other approaches to this are formal power series and divergent integrals (your example would be represented as $\int_0^\infty dx$).
In the theory of divergent integrals or the theory of hyperfunctions (see Urs Graf) one can show that $\int_0^\infty dx=\int_0^\infty \frac1{x^2} dx$, thus indeed the "germ" of function $f(x)=x$ at infinity in a sense is equal to the germ of $g(x)=1/x$ at zero (from positive direction).
Following Urs Graf's "Introduction to Hyperfunctions and Their Integral Transforms", page 36:
If we insert here $x=0$, we will get $-i\pi\delta(0)$, which is the "value" of $\frac1{i0}$, multiplying the both sides by $i$ we get $1/{0^+}=\pi\delta(0)$, which is equal to the half of Fourier transform of $f(x)=1$ (which is $\int _{-\infty }^{\infty }f(x)e^{-i\nu x}\,dx=2\pi\delta(\nu)$) evaluated at $\nu=0$, that is, $\int_0^\infty1dx=\pi\delta(0)=\frac{1}{0^+}$
Notice though that we cannot ascribe this value to $1/0$ because the function $1/x$ has germs at zero of opposite signs from the positive and negative directions.
The constant which you denote as $\infty_0$ is usually denoted as $\omega$ or $\varepsilon^{-1}$ (for instance, in Levi-Civita field, which is a kind of formal power series).