= Solution
For $T>T_B$, the ground-state occupation is negligible to leading order, so $N\simeq N_\epsilon$. Comparing the number equation with its value at $T_B$ gives
$$
\frac{\epsilon-\mu}{T}=\frac\epsilon{T_B},
\qquad
\boxed{\mu=\epsilon\frac{T_B-T}{T_B}.}
$$
For $T<T_B$, $\mu$ is close to zero and
$$
N_0\simeq-\frac{kT}{\mu},
\qquad
N_\epsilon\simeq\frac{M}{e^{\epsilon/(kT)}-1}.
$$
Using $M=N(e^{\epsilon/(kT_B)}-1)$ in $N_0=N-N_\epsilon$ and solving for $\mu$ gives
$$
\boxed{\mu\simeq-\frac{kT}{N}
\frac{e^{\epsilon/(kT)}-1}
{e^{\epsilon/(kT)}-e^{\epsilon/(kT_B)}}.}
$$
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