= Solution
Put $q=10/Re>0$. The frequency formula becomes
$$
\omega_\pm=\omega_1-\frac q2\pm i\sqrt{q-\frac{q^2}{4}}.
$$
For large positive <Reynolds number>, $\sqrt{q-q^2/4}=\sqrt q[1-q/8+O(q^2)]$. Therefore
$$
\boxed{\omega_\pm=\omega_1\pm i\sqrt{\frac{10}{Re}}+O(Re^{-1}).}
$$
More precisely the real correction is $-5/Re$, and the next imaginary correction is $O(Re^{-3/2})$. The growing branch has a rate tending to zero as $Re\to\infty$, despite being unstable at every sufficiently large finite $Re$.
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