Why can't we determine the limit of $\cos x$ and $\sin x$ at $x=\infty $ or $x=-\infty$? I'm confused about why we can't determine the limit of $\cos x$ and $\sin x$ as  $x \to \infty$, even though they are defined over  $\mathbb{R}.$
When I use Wolfram Alpha, I get the following result (link to page):



which shows only that there are $2$ limits :$-1$ and $ 1 $.
Can someone show me why we can't determine $\lim \sin x$ and $\lim \cos x$ at  $x=\infty $ or $x=-\infty$ ?
Thank you for your help.

They can't have a limit because they're periodic functions. What Wolfram Alpha outputs are the limit inferior and the limit superior of these functions, which always exist as soon as the functions are bounded. In case you haven't
Similarly seen these notions yet, by definition:
$$\limsup_{x\to\infty}f(x)=\lim_{x\to\infty}\sup_{t\ge x}f(t)$$
Note that the function $g(x)=\sup_{t\ge x}f(t)$ is non-increasing, hence it has a limit if $f$  is bounded from below.
Similarly,
$$\liminf_{x\to\infty}f(x)=\lim_{x\to\infty}\inf_{t\ge x}f(t)$$
which exists because the function $h(x)=\inf_{t\ge x}f(t)$ is non-decreasing, hence it has a limit if $f$  is bounded from above.
Also, one can prove that for any number $a\in[-1,1]$ there exists a sequence $(x_1, \dots, x_n,\dots) \to \infty$ such that $(f(x_1),\dots,f(x_n),\dots)\to a$.
