Can asymptotes be curved?
When I was first introduced to the idea of an asymptote, I was taught about horizontal asymptotes (of form y=a) and vertical ones ( of form x=b).
I was then shown oblique asymptotes-- slanted asymptotes which are not constant (of the form y=ax+b).
What happens, though, if we've got a function such as
Is considered an asymptote in this example?
Another example, just to show you where I'm coming from, is
-- is an asymptote in this case?
The reason that I ask is that I don't really see the point in defining oblique asymptotes and not curved ones; surely, if we want to know the behaviour of y as , we should include all types of functions as asymptotes.
If asymptotes cannot be curves, then why arbitrarily restrict asymptotes to lines?