I wonder whether anyone ever tried to introduce an extension of real numbers by adding an element $\nu$ which would signify the behavior of the function $(-1)^x$ as $x$ goes to infinity?

In other words, in some generalization of limit, $$\operatorname{gen}\lim_{x\to\infty} (-1)^x =\nu$$

This value $\nu$ should characterize the particular way in which $(-1)^x$ behaves at infinity, so that if a function oscillates at infinity in a different way, for instance, with different frequency, it has a different "limit" but it still may be expressable in terms of $\nu$, if we can do some operations on the function to make it behave like $(-1)^x$ at infinity. For instance, $\operatorname{gen} \lim_{x\to\infty} \cos^2 (\pi x/2)=(\nu+\frac1\nu)/2+1/2$. I think, due to Fourier series we can represent the behavior of any periodic function at infinity in terms of $\nu$, though the series could turn infinite.

Introduction of such element would also allow to represent the series like $\sum_{k=0}^\infty (-1)^k$ more precisely than just giving the Cesaro-regularized value.

The functions like $\cos (1/x)$ can be made continous at zero...

Particularly we can define

$$\operatorname{gen}\lim_{x\to\infty} \cos ax = \frac{\nu^{a/\pi}+\nu^{-a/\pi}}2$$

and

$$\operatorname{gen}\lim_{x\to\infty} \sin ax = i\frac{\nu^{-a/\pi}-\nu^{a/\pi}}2$$

This way sine and cosine would obey the Pythagorean trigonometric identity even at infinity, despite the Cesaro-regularized (mean) value of the both is zero.

Moreover, we can represent some non-periodic functions this way as well, for instance we would formally have

$$\operatorname{gen}\lim_{x\to\infty} \exp x = \nu^{-i/\pi}$$

We also would be able to define Dirac Delta function as an elementary function: $$\delta (z)=\frac{\nu ^{z/\pi }-\nu ^{-z/\pi }}{2 \pi i z}$$

That said, I wonder whether anyone came up with an idea of oscillating unity, and what would be the algebraic properties of such "number"?