= Solution
Holding emissions at $E_{00}=8.069\,\mathrm{GtC\,yr^{-1}}$ gives equilibrium concentration
$$
C_\infty=C_0+\frac{b_1E_{00}}{b_0}\simeq682.43\,\mathrm{ppmv}.
$$
For time $t$ since 2000,
$$
\boxed{C(t)=C_\infty+(C_{00}-C_\infty)e^{-t/(200\,\mathrm{yr})}.}
$$
It increases monotonically on the same 200-year timescale and reaches about $654.07\,\mathrm{ppmv}$ in 2500. The limiting, and hence peak asymptotic, warming is
$$
\boxed{\Delta T_{\rm peak}\longrightarrow3\log_2(682.43/279)\simeq3.87\,\mathrm K.}
$$
In 2500 the equilibrium warming is about $3.69\,\mathrm K$, still below that asymptote. A stable thermal adjustment much faster than the carbon timescale follows this slow approach with a short lag and has the same limiting warming. These are predictions of the specified linear reservoir model, not a claim that constant real emissions produce a finite universal atmospheric equilibrium.
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