I used this idea in Algebra 2 during our polynomial graph sketching unit to support their conceptual understanding of end behavior, and it worked surprisingly well.
Most of my students had not had much truck with the idea of infinity, much less with the idea of approaching it, so we started by talking about how the graph of a function is a representation of a specific story — like a novel — or like the life story of a human being. Every person's life story is completely unique, yet everyone has an experience of both birth and death (the beginning and end being like end behavior). Everyone's birth story may be different -- some people are born in a taxi cab, some are born at home, some are born in a hospital, some are born under water, and yet every single one of these births has the very fact of having gone through birth in common. Similarly, death is the ultimate "end behavior" of life. We cannot know exactly how or when a specific person's death may come, but we do know that it is going to happen some time after they have been born and have lived for some duration of time.
We also talked about the movement of a function graph across the axis at its x-intercepts — how at certain key transformative moments, a person's life crosses from negative into positive territory -- or it may move in the opposite direction.