Let $\{(x,y)\}_{x\in\mathbb R}$ be a continuous curve; what will be implied if it is given that the any rotated version of that curve have a limit?

i.e. let $(s,t)=A(x,y)$ be a rotation around the origin, $\lim_{s\to+\infty}(s,t)$ exists or tends to infinity.

We say the mapping $f$ has a limit if the limit exists or tend to $\infty$. i.e. $\lim_{x\to +\infty}f(x)=K\subset [-\infty,+\infty]$ where $K$ is a finte set, $f$ is a correspondence.

First Case: the limit of a curve exists, and, the limit of the rotated curves also always exist. For example, the limit of the curve $\lim_{x\to+\infty}(x,x)=(+\infty,+\infty)$ exists, and the limit of $\{(x,ax)\}_{x\in\mathbb R}$ also exist for any $a$. In fact, the limit of any polynomial curve $\{(x,f(x))\}_{x\in\mathbb R}$ (where $f(x)=\sum_n a_nx^n)$ exist and so do their rotated point sets. We call that the curve "has a limit in every direction".

Second Case: Sometimes, the limit of a curve does not exist, but if rotate the curve, then the limit might exist. For example, the curve $\{(x,\sin(x))\}_{x\in\mathbb R}$ does not have a limit, but rotated curve $\{(s,t)\}_{s\in\mathbb R}$, where $$(s,t)=(\frac{1}{\sqrt 2}(x-\sin(x)),\frac{1}{\sqrt 2}(x+\sin(x))),$$ does have a limit: $\lim_{s\to +\infty}(s,t)=(+\infty,+\infty)$

Question: if we know that the curve {(x,y)} has limit in every direction, what can be implied from this kind of convergence? Does that mean that the function is asymptotically polynomial? Or, asymptotically not a periodic function?


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