I encountered the limit with the form $$ \lim_{r\rightarrow 1^+}\frac{1-\frac{1}{r^{d-3}}}{e^{2\kappa r_s(r)}},\quad r_s(r)\equiv r~ F_1\left(-\frac{1}{d-3};-\frac12,1;\frac{d-4}{d-3},-\frac{Q}{r^{d-3}},\frac{1}{r^{d-3}} \right) $$ where $r>0$, $F_1$ is the Appell F1 function, $\kappa=\frac12\frac{1}{\sqrt{1+Q}}$, $d=5,6,7,\cdots$ and $Q\geq0$.
Question: What is the value of this limit when $Q > 0$?
What I tried so far:
When $Q\neq0$, Mathematica gives zero, which is surely wrong as I tested numerically.
I can find the limit in the case of $Q=0$. When $Q=0$, the F1 function reduces to Hypergeometric function $$ F_1\left(-\frac{1}{d-3};-\frac12,1;\frac{d-4}{d-3},0,\frac{1}{r^{d-3}} \right)= {}_2F_1\left( 1,\frac{1}{3-d},\frac{1}{3-d}+1,r^{3-d} \right) $$ and the limit is given by Mathematica as $$ \exp\left[{-\gamma_E-\psi^{(0)}\left(\frac{1}{3-d}\right)}\right] $$ where $\gamma_E$ is the Euler gamma and $\psi^{(0)}$ is the polygamma.