Using convergence theorem without constructing a sequence of function

Question

Denote $\hat{f}$ be the Fourier transformation of $f$ and both $f, \hat{f} \in L^1(\mathbb{R})$ are continuous. In a proof of Fourier inversion Theorem for $L^1$ function, , I have seen a trick as following $$\int \hat{f}(\xi) e^{i\xi x}d\xi = \lim_{\epsilon \to 0}\int \hat{f}(\xi) e^{\epsilon \xi^2} e^{i\xi x}d\xi$$ I am getting confusion about the operation of moving limit at here. I know the Dominated Convergence Theorem or Monotone Convergence Theorem for some sequence of function $\{f_n\}$, but in the trick present above, they taking limit as $\epsilon \to \infty$ other than construct a sequence of function, so would this also be an application of any Convergence theorem? Thanks for any explanation.