Let $\psi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ such that, for all $x \in \mathbb{R}$,
$$\psi(x) = \begin{cases} \dfrac{x}{\mathrm{e}^{x} - 1} & \text{if } x \neq 0 \\ 1 & \text{otherwise} \end{cases}$$
Let furthermore $u$ be the function from $\mathbb{R}^{2}$ to $\mathbb{R}$ such that, for all $(x,t) \in \mathbb{R}^{2}$,
$$u(x,t) = \psi(x)\,\mathrm{e}^{tx}.$$
Show that $u$ is of class $\mathcal{C}^{\infty}$ on $\mathbb{R}^{2}$.