Most students come to calculus with an intuitive sense of what a tangent line should be for a curve. It is easy enough to give a definition of a tangent to a circle that is both elementary and rigorous. (A line that intersects a circle exactly once.) Yet, when we talk about a curve, such as a polynomial, I think one must talk about infinity to give a rigorous definition. This is not a bad thing... It can help motivate a lesson on the formal definition of a tangent line at a point... But, what intermediate definition could I use to help those students who have no intuition about the matter get an idea of what we are going for before we talk about secants and such?
I have looked at some textbooks and online and I find saying a tangent line is one that "just touches" problematic-- it is the kind of phrase that is quite meaningless... Even in the setting of un-rigorous definitions. Yes, I will give examples, but I would like to do better ... I think saying that a tangent line is an arrow that points in the direction that the curve is going at that instant might make sense... Though it makes everything 'directional' and that might confuse them later.
What is the best intermediate definition you have seen?