= Solution
Using mass $7\,\mathrm u$ for the dominant <lithium> isotope gives a transition <wavenumber> $\widetilde\nu_0=14925\ \mathrm{cm^{-1}}$. Inserting the thermal speed into the stated <Doppler broadening> expression gives
$$
\Delta\widetilde\nu_D={\widetilde\nu_0\over c}\sqrt{\frac{2k_BT\ln2}{m}}
\simeq0.128\ \mathrm{cm^{-1}}.
$$
The supplied formula is the Gaussian <half width at half maximum>; the corresponding full width is about $0.256\ \mathrm{cm^{-1}}$. Meanwhile exponential population decay with lifetime $\tau$ produces a <natural linewidth> expressed as a full width
$$
\delta\widetilde\nu_{\rm nat}=\frac{1}{2\pi c\tau}\simeq1.97\times10^{-4}\ \mathrm{cm^{-1}}.
$$
Here $c$ must be in $\mathrm{cm\,s^{-1}}$ in the second expression. Equivalently the fine-structure separation is about $9.0$ GHz, the Doppler half width about $3.83$ GHz, and the <natural linewidth> expressed as a full width about $5.89$ MHz. The <natural linewidth> is therefore negligible compared with thermal broadening; low density also makes collision broadening less important.
\b[The doublet is resolvable in principle, but only marginally against the Doppler width.] The separation exceeds one Doppler full width, so two components can survive, although their profiles overlap strongly. The $J=3/2$ and $J=1/2$ upper levels have different <statistical weights>, so the weaker member need not make an equally conspicuous maximum. A <spectral resolving power> $R\gtrsim\widetilde\nu_0/0.3\simeq5\times10^4$ is the basic instrumental requirement; a width well below $0.1\ \mathrm{cm^{-1}}$ and adequate signal would make the test more convincing. An arbitrarily fine spectrometer cannot remove the physical Doppler overlap. This conclusion uses the stipulated thermal, optically thin picture; additional photospheric velocity fields or optical-depth effects would require more information.
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