Biroli et al.'s extension of the standard mode-coupling theory to inhomogeneous equilibrium states [Phys. Rev. Lett. 97, 195701 (2006)] allowed them to identify a characteristic length scale that diverges upon approaching the mode-coupling transition. We present a numerical investigation of this length scale. To this end we derive and numerically solve equations of motion for coefficients in the small q expansion of the dynamic susceptibility χ_q(k;t) that describes the change of the system's

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... namics due to an external inhomogeneous potential. We study the dependence of the characteristic length scale on time, wave-vector, and on the distance from the mode-coupling transition. We verify scaling predictions of Biroli et al. In addition, we find that the numerical value of the diverging length scale qualitatively agrees with lengths obtained from four-point correlation functions. We show that the diverging length scale has very weak k dependence, which contrasts with very strong k dependence of the q→ 0 limit of the susceptibility, χ_q=0(k;t). Finally, we compare the diverging length obtained from the small q expansion to that resulting from an isotropic approximation applied to the equation of motion for the dynamic susceptibility χ_q(k;t).