I am not sure from where you are getting these arguments, but they are somewhat backwards. The whistler wave is the high frequency branch of the dispersion relation in both a hot and cold plasma.
The dispersion relation you show is not actually the fast/magnetosonic mode or whistler mode dispersion relation. It's a very specific limit where the waves are only allowed to propagate parallel to the quasi-static magnetic field. The $Z\left( \zeta \right)$ is the plasma dispersion function given by:
$$
Z\left( \zeta \right) = \frac{ 1 }{ \sqrt{ \pi } } \int_{C} \ dz \ \frac{ e^{-z^{2}} }{ z - \zeta } \tag{1}
$$
where the contour, $C$, is understood to follow the real z-axis and pass under the pole at $z = \zeta$.
In the limit of small k (i.e., large $\zeta$), you would use the Sokhotski–Plemelj theorem and a Taylor expansion to show that Equation 1 goes to:
$$
Z\left( \zeta \right) \simeq i \frac{ k }{ \lvert k \rvert } \sqrt{ \pi } \ e^{- \zeta^{2}} - \left[ \frac{ 1 }{ \zeta } + \frac{ 1 }{ 2 \zeta^{3} } + \frac{ 3 }{ 4 \zeta^{5} } + ... \right] \tag{2}
$$
Note that we have used $\zeta = x + i y$ here, where $x = \tfrac{ \omega }{ V_{Te} k }$, $y = \tfrac{ \gamma }{ V_{Te} k }$, $\omega$ is the real part of the frequency, and $\gamma$ is the imaginary part. Here we note that $V_{Te}$ is the most probable thermal speed for electrons.
In the limit of large k (i.e., small $\zeta$), you need to make a substition first to get a linear differential equation. The solution is found in any math formulary book. After expanding power series about small $\zeta$, one can show that:
$$
Z\left( \zeta \right) \simeq i \frac{ k }{ \lvert k \rvert } \sqrt{ \pi } \ e^{- \zeta^{2}} - 2 \zeta + \frac{ 4 }{ 3 } \zeta^{3} - \frac{ 8 }{ 15 } \zeta^{5} + ... \tag{3}
$$
So you can see from Equation 3 that in the limit as $\zeta \rightarrow 0$, we find that $Z\left( \zeta \right) \rightarrow 0$ as well. If we limit ourselves to just your dispersion relation, then the result will be a parallel propagating light wave in vacuum.
