The asymptoticity of extremal length in Teichm\"uller space

We study the asymptotic behavior of extremal length along Teichm\"uller rays. Specifically, we determine the limit of extremal length along a Teichm\"uller ray and obtain an explicit expression for this limit, which complements a related formula established by Cormac Walsh. Building on this result and Kerckhoff's formula, we establish a formula for the limiting Teichm\"uller distance between two points moving along arbitrary pairs of Teichm\"uller rays. Furthermore, we derive a necessary and sufficient condition for two Teichm\"uller rays to be asymptotic. Finally, by shifting the initial points of the Teichm\"uller rays along their associated Teichm\"uller geodesics, we show that the minimum of the limiting Teichm\"uller distance coincides with the detour metric between the endpoints of the rays on the horofunction boundary.