Were the graph of the asymptotic function $n/x$ rotated counterclockwise about the origin $45^O$, its derivation point would be 0 at $\sqrt{n}$ rotated similarly about the y axis where $n/x=x$.
The inflection point has significance, viz the derivative is 0. Does the point where the asymptote curves have any significance? $n=1$ 
. Visually the point has significance but I'm wondering if it has mathematical significance.
Answering this question as well as some links to relevant reading would be much appreciated.
In general the point where $ f(x) = x $ is called the fixed point. However what I think the term you are looking for is the "vertex" of the hyperbola.
This is the point where the two branches are the nearest to each other and is where a line draw between the two foci of the hyperbola crosses the hyperbola.