  Phase transitions and critical phenomena, which are dominated by fluctuations
and correlations, are one of the fields replete with physical paradigms and
unexpected discoveries. Especially for two-dimensional magnetism, the
limitation of the Ginzburg criterion leads to enhanced fluctuations breaking
down the mean-field theory near a critical point. Here, by means of magnetic
resonance, we investigate the behavior of critical fluctuations in the
two-dimensional ferromagnetic insulators $\rm CrXTe_3 (X=Si, Ge)$. After
deriving the classical and quantum models of magnetic resonance, we deem the
dramatic anisotropic shift of the measured $g$ factor to originate from
fluctuations with anisotropic interactions. The deduction of the $g$ factor
behind the fluctuations is consistent with the spin-only state (${g\approx}$
2.050(10) for $\rm CrSiTe_3$ and 2.039(10) for $\rm CrGeTe_3$). Furthermore,
the abnormal enhancement of $g$ shift, supplemented by specific heat and
magnetometry measurements, suggests that $\rm CrSiTe_3$ exhibits a more typical
two-dimensional nature than $\rm CrGeTe_3$ and may be closer to the quantum
critical point.
