I'm attempting to identify the type of discontinuity at $(x,t)=(0,0)$ for
$$u(x,t)=\frac{x}{t^{3/2}}e^{\frac{-x^2}{4t}}$$
My gut tells me it's a removable discontinuity, but I don't see anything immediately obvious because the fractions are like $0\times\infty$. I know in $1$-dimension the type of discontinuity can be identified by taking limits around the discontinuity, but I'm unsure how to proceed with this type of analysis in multiple dimensions.
For context, $u(x,t)$ is a solution of the Diffusion IBVP:
\begin{cases} u_t=u_{xx}, & \text{for $0<x<\infty$, $t>0$} \\ u(0,t)=0, & \text{for $t>0$} \\ u(x,0)=0, & \text{for $x\ge0$} \end{cases}
Hint: What happens when $t = x^2$?