I have a function $f(x,y,z) :\mathbb{R}^3 \rightarrow \mathbb{C}$, a smooth function. I know that $$ I = \int_{z \in \mathbb{R}}\int_{y \in \mathbb{R}}\int_{x \in \mathbb{R}} f(x,y,z) \ dx dydz $$ this integral converges. However, it does not converge absolutely, i.e. $$ \int_{z \in \mathbb{R}}\int_{y \in \mathbb{R}}\int_{x \in \mathbb{R}} |f(x,y,z)| \ dx dydz = \infty. $$
My question is the following: Does the following limit always hold? $$ I = \lim_{\varepsilon \rightarrow 0} \int_{-L_1}^{L_1} \int_{-L_2}^{L_2} \int_{-L_3}^{L_3} e^{- \varepsilon x^2}f(x,y,z) \ dx dydz, $$ where $L_1, L_2, L_3$ grows to infinity as $\varepsilon$ goes to $0$.
If it was absolutely convergent then it is clear from the dominated convergence theorem. But I wasn't sure about this case... I would appreciate any comments, hints and suggestions. Thank you very much.