Anyone who has taught mathematics has inevitably run into the confusion students make between tangent and asymptote. And yet it is not for lack of saying, writing, repeating, and copying out that a curve approaches its asymptote without ever reaching it, while a tangent touches the curve!
One concerns points at a finite distance, the other points located at infinity. Yes, but when we draw curves, we never go all the way to infinity! There is nevertheless a trick for "seeing" what happens at infinity. Gain some height, and look at the graph in perspective (think of how a pilot sees the runway on takeoff).
We shall see that asymptotes become tangents.
Constructing a perspective view
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Let's indeed gain some height. Let's add a third dimension (Oz) to the plane (x O y) of the figure below, and place the eyes of an observer at Ω, with coordinates
(0, –d, h) where h is the height above the "ground" (xOy) and d is the distance to the "screen" (x O z*).
