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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?

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?

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 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?

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?

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 break the main trigonometric identity $\sin^2 x+\cos^2 x =1$ at this point).

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?

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 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?