= Solution
Let $a$ be a stable equilibrium separation: $V'(a)=0$, $K=V''(a)>0$. With bond extension $\delta=x-a$, the <Taylor expansion> is
$$
V(a+\delta)=V(a)+\tfrac12K\delta^2+\tfrac16V'''(a)\delta^3+\cdots.
$$
The <harmonic approximation> applies when the typical thermal and zero-point bond fluctuations satisfy $|\delta|\ll a$ and the higher terms are negligible, for example $|V'''(a)\delta|\ll3K$ and $|V''''(a)\delta^2|\ll12K$. The relevant small quantity is the relative displacement of neighbouring atoms. \b[Small bond excursions about a stable minimum give an amplitude-independent <spring constant> $K=V''(a)$.] Large thermal excursions, bond breaking or appreciable <anharmonicity> invalidate the approximation.
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