= Solution
Put $Q=R_QT$ and $B=R_BT$ in the root-mean-square accuracy measure:
$$
Z(T)=\frac{R_Q\sqrt T}{\sqrt{R_Q+(1+f)R_B+(1-f)^2R_B^2T}}.
$$
For a fixed uncorrected $f\neq1$ and $R_B>0$, the squared bias eventually dominates the shot-noise terms. Therefore
$$
\boxed{\lim_{T\to\infty}Z(T)=\frac{R_Q}{|1-f|R_B}.}
$$
The PDF omits the absolute value. Its expression is the positive accuracy ratio only if $f<1$; for $f>1$ the residual background changes sign, but an error magnitude and a <signal-to-noise ratio in photon counting> remain nonnegative. The printed signed formula can instead be read as source divided by signed bias.
For exact background matching $f=1$, there is no <systematic-error signal-to-noise ceiling>: $Z=R_Q\sqrt{T/(R_Q+2R_B)}$ grows without bound in this idealized model. With $R_B=0$ the same conclusion holds. Limits in which $f$ itself approaches one with exposure time are different from the fixed-mismatch limit used here.
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