TITLE: Rigorous way to find the limit of this difference?
QUESTION [9 upvotes]: This is a question from an old released exam.

By the triangle inequality, $s-r<1$, so I eliminate answers D and E. Intuitively, since the lower angle between $1$ and $r$ is fixed at $110^\circ$, $s$ will always be a little longer than $r$, so I eliminate A and C to find B as the correct answer.
This is pretty informal, is there a more rigorous way one could prove the limit?

REPLY [2 votes]: A qualitative answer.
When $s$ and $r$ tend both to infinity, keeping the said elements fixed, the sides $s$ and $r$ tend to be parallel, so their difference is the projection of the third side on their common direction, given by
$$s-r\to|1\cdot\cos\alpha|$$