I usually level with my students about this, but with the caveat that any serious use of infinity involves a precise and technical definition, in particular one

I usually level with my students about this, but with the caveat that any serious use of infinity involves a precise and technical definition, in particular one that is not guessed on the fly as students tend to do. To illustrate, I then talk about how in elementary calculus infinity is a precise and technical shorthand for the idea of unbounded growth. [Edit:] I also recently had the wonderful opportunity to talk to really advanced high school students about projective geometry, and I contrasted the difference between the way people talk about infinity (the point on the horizon! I'm just making it up!) versus the precise and technical definitions (a specific point in a quotient of a vector space).