It is informal defination. Truly, it confuses a lot if you are not patient enough to grasp the idea.
At this time, I was unable to delete the message. On other time, I might resend it with clear and formal way of prooving it. What I prsented there is , "Informal way of prooving". I appolgise for that. I on my way, was trying to make my idea very easy but the outcome is unattractive.
What you wrote looks OK to me but it's definitely not what I'm used to seeing in a proof (you're right it was very "informal") and I have a few holes to fill in (pun intended). You say that asymptotes are *the* restriction of the domain of the domain/range of a function. You also say that if the numerator of f(x) = 0 then f(x) has no asymptotes and, by your definition, no restrictions in the domain. What if the denominator is a polynomial of x of degree greater than 0? For example, f(x) = 0/(x-1) is not defined at x = 1 and therefore has a domain restriction (but you're right it isn't an asymptote) so I think you may want to refine your alternate definition of an asymptote a little. I like most of the rest of it though.