A function's limit gives infinity symbol with tilde I made a function while studying some unsolved problem ($3x+1$ problem) and wanted to find a function's limit. Im no well educated mathematician so I used Symbolab with L'Hopital's rule. I got this strange infinity symbol:
$$\lim_{x \to \infty} f(x) = \tilde{\infty}$$
What does the infinity with the tilde above symbol means? (in context to limits).
I read that some function approaches this complex infinity. But, what do one do when one gets such an result. Is it related to the Riemann Sphere? Is the Collatz Conjecture related to the Riemann Sphere, or am I out on a blueberry trip?
 A: The symbol $\tilde\infty$ is used, somewhat confusingly, for a few different things in different contexts. It can mean:

*

*the point at infinity on the Riemann sphere (or one-point compactification of $\Bbb R$ in contexts where $\infty$ would otherwise denote "positive infinity")


*an unspecified directed infinity in the complex plane (or in $\Bbb R^n$, if you're a terrible person)


*a directed infinity in the complex plane whose real part is nonnegative (I've only seen this once, and I don't think it's common)
I'm no math historian, but I've seen older textbooks use $\tilde{(\cdot)}$ to indicate that a variable represents a complex number. Based on that, I would guess that $\tilde\infty$ was used - at least at some point - to indicate divergence of complex valued functions (as opposed to the more familiar $-\infty,+\infty$ that appears in e.g. Rudin's Principles of Mathematical Analysis).
As for what $\tilde\infty$ means in Symbolab, specifically: I have no idea. I don't know how Symbolab works, and there doesn't seem to be any kind of documentation.
