Let $\kappa>0$, let $s>n/2+2$, and let $u_0\in H^{s+2}(\mathbb R^n)$. For each $\tau>0$, let \begin{align*} (u_0^\tau,u_1^\tau)\in H^{s+2}(\mathbb R^n)\times H^s(\mathbb R^n) \end{align*} satisfy \begin{align*} \|u_0^\tau-u_0\|_{H^{s+2}(\mathbb R^n)}\to0 \end{align*} and \begin{align*} \|u_1^\tau-\kappa\Delta u_0\|_{H^s(\mathbb R^n)}\to0 \end{align*} as $\tau\downarrow0$. Let \begin{align*} u^\tau:[0,\infty)\times\mathbb R^n\to\mathbb R \end{align*} be the Fourier-semigroup solution of \begin{align*} \tau u_{tt}^\tau+u_t^\tau-\kappa\Delta u^\tau=0 \end{align*} with $u^\tau(0)=u_0^\tau$ and $u_t^\tau(0)=u_1^\tau$. Let \begin{align*} v:[0,\infty)\times\mathbb R^n\to\mathbb R \end{align*} be the Fourier-semigroup solution of the [heat equation](/page/Heat%20Equation) \begin{align*} v_t-\kappa\Delta v=0,\qquad v(0)=u_0. \end{align*} Then, for every $0<\delta<T<\infty$, \begin{align*} \sup_{\delta\le t\le T}\|u^\tau(t)-v(t)\|_{H^s(\mathbb R^n)}\to0 \end{align*} as $\tau\downarrow0$.