Let's consider the affinely extended real line. The functions that have a limit on positive or negative infinity $\lim_{x\to+\infty} f(x)$ or $\lim_{x\to-\infty} f(x)$ can be generalized to the values at infinity $f(+\infty)$, $f(-\infty)$ based on these limits.

But what about functions that do not have such limits?

For instance, if we take these integrals following Borel generalization, we get:

$\int_0^\infty \cos x=0$

and

$\int_0^\infty \sin x=1$

Since integral of $\cos x$ is $\sin x$ and integral of $\sin x$ is $-\cos x$, we can probably generalize these functions to infinity (assuming the fundamental theorem of calculus should still hold): $\sin (\infty)=\cos(\infty)=0$ (notice, this would apparently break the main trigonometric identity $\sin^2 x+\cos^2 x =1$ at this point but one of the books linked in the comments resolves the paradox by claiming that $(\sin \infty)^2\ne \sin^2 \infty$ and the same for cosine).

Still, I wonder whether there were other attempts at such generalization? Possibly, something along Cesaro lines (limit of mean value of the function) or sometyhing else?