a limiting parallel
In the hyperbolic plane, 'parallel' splits into two genuinely different relationships, and the limiting parallel is the special borderline case between them. Fix a line L and a point P off it. Among all the lines through P that never meet L, two are distinguished: as you rotate a ray about P from the direction that does hit L toward the direction that does not, there is a precise first ray that just barely fails to touch L. There is one such borderline ray on each side, and these two are the limiting parallels to L through P. Every line strictly between them is an 'ultraparallel' that diverges from L in both directions.
What makes a limiting parallel special is its asymptotic behaviour. A limiting parallel never meets L, yet it gets arbitrarily CLOSE to L as you travel along it in one direction — the gap between them shrinks toward 0 without ever closing. In the other direction the two lines pull apart. This is utterly unlike Euclidean parallels, which stay a fixed distance apart forever. The angle the limiting parallel makes with the perpendicular from P to L is exactly the angle of parallelism, Pi(p), set by the distance p alone. The ultraparallels, by contrast, share a unique common perpendicular and recede from L on both sides.
So the hyperbolic plane has TWO flavours of non-intersecting line: the two limiting (asymptotic) parallels that hug L at infinity, and the infinitely many ultraparallels that veer away. When people say a point in the hyperbolic plane has 'infinitely many parallels' to a line, both kinds are being counted. The honest subtlety: limiting parallelism is direction-dependent — a line is limiting-parallel to L in a specific sense of travel — which is why the careful literature distinguishes the two limiting parallels from the divergent rest.
In the Poincare disk, model L as a diameter and P as a point above it. Two arcs through P run to the very endpoints of L on the boundary circle — these are the limiting parallels, meeting L only 'at infinity' (on the rim). Any arc through P that ends strictly between those two endpoints is an ultraparallel; it shares one common perpendicular with L and drifts away on both sides.
The two borderline rays that just miss L are the limiting parallels; everything strictly between them diverges from L.
It is a mistake to treat all hyperbolic 'parallels' as one kind. Limiting (asymptotic) parallels approach a line without meeting it; ultraparallels share a common perpendicular and diverge — only the former realise the angle of parallelism.