Why are Complexity Notations Called Asymptotic?
I would like to quote from &quot;Concrete Mathematics&quot; (Chapter 9) by Ronald Graham, Donald Knuth, and Oren Patashnik. It does mention curves and asymptotes.

The word asymptotic stems from a Greek root meaning &quot;not falling together&quot;. When ancient Greek mathematicians studied conic sections, they considered hyperbolas like the graph of $y = \sqrt{1 + x^2}$ which has the lines $y = x$ and $y = -x$  as &quot;asymptotes&quot;. The curve approaches but never quite touches these asymptotes, when $x \to \infty$.
Nowadays we use &quot;asymptotic&quot; in a broader sense to mean any approximate value that gets closer and closer to the truth, when some parameter approaches a limiting value [emphasis added]. For us, asymptotics means &quot;almost falling together&quot;.