I was having trouble getting my students at IGCSE level to remember what an asymptote is so I made up a quick analogy which worked well.
I told them to think of a situation in which they were friends with someone that they fancied but the feeling wasn´t reciprocated. As time goes by, they become better and better friends with that person but unfortunately they never actually get the opportunity to go on a date. It’s the same with curves and asymptotes. The curve gets closer and closer to the asymptote but never gets the opportunity to date it.
(See my Geogebra applet on Vertical Asymptotes and the Domain of a Function)