Are there any known approaches of generalizing functions that do not have a limit at infinity to values at infinity?

Question

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 (or Abel) 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?

Answer 1

In physics we use the Sokhotski–Plemelj theorem to evaluate integrals from zero to infinity of cosine and sine in the form:

$$F(k)=\int_0^\infty (\cos kx +i\sin kx)\,dx=\pi\delta(k)+i\,{\cal P}\frac{1}{k}$$

where $\delta(k)$ is the Dirac delta function and ${\cal P}$ is a reminder that when the right-hand-side is integrated over $k$ one should take the Cauchy principal value of the integral: $$\int_{-\infty}^\infty F(k)g(k)\,dk=\pi g(0)+i\,{\cal P}\,\int_{-\infty}^\infty \frac{g(k)}{k}\,dk.$$

Incidentally, from $F(1)=i$ I would indeed associate $\int_0^\infty \cos x\,dx$ and $\int_0^\infty \sin x\,dx$ with $0$ and $1$, respectively, as in the OP.

Answer 2

Many people developed a theory of limits of distributions at finite or infinite values in the 50's and 60's and for these all of your formulae hold. A particularly elementary version was developed by J. Sebastião e Silva and a complete description can be found online by googling has name and "limits of distributions at infinity". There is also a text book version by J. Campos Ferreira et al.---"Introduction to the theory of distributions".