I would like to phrase properly the following result.
Let $f(x,\mathbf{y})$ be some bounded and continuous function for all $x\in\mathbb{R}$ and $\mathbf{y}=(y_1,...,y_k)^T\in\mathbb{R}^k$ is a vector of variables, such that there exist a function $w(x)$ such that $\frac{f(x,\mathbf{y})}{w(x)}$ is bounded for all $x\in\mathbb{R}$ and $\mathbf{y}\in\mathbb{R}^k$ and $\lim_{|x|\to\infty} \frac{f(x,\mathbf{y})}{w(x)}=\psi(\mathbf{y})$. Now consider the expression \begin{eqnarray*} h(\mathbf{y}) & = &\lim_{|x|\to\infty}\frac{f(x,\mathbf{y})\phi(\mathbf{y})}{\int_{\mathbf{Y}}f(x,\mathbf{y})\phi(\mathbf{y})d\mathbf{y}}=\lim_{|x|\to\infty}\frac{\frac{f(x,\mathbf{y})}{w(x)}\phi(\mathbf{y})}{\int_{\mathbf{Y}}\frac{f(x,\mathbf{y})}{w(x)}\phi(\mathbf{y})d\mathbf{y}}=\frac{\psi(\mathbf{y})\phi(\mathbf{y})}{\lim_{|x|\to\infty}\int_{\mathbf{Y}}\frac{f(x,\mathbf{y})}{w(x)}\phi(\mathbf{y})d\mathbf{y}}\\ & = & \frac{\psi(\mathbf{y})\phi(\mathbf{y})}{\lim_{|x|\to\infty}\int_{Y_1}...\int_{Y_k}\frac{f(x,(y_1,...,y_k))}{w(x)}\phi(\mathbf{y})dy_k...dy_1}. \end{eqnarray*} Note that we have $k$ integrals and $Y_1,...,Y_k$ are the support of $y_1,...,y_k$, respectively, hence ${\mathbf Y}={\mathbf Y_1}\times...\times{\mathbf Y_k}$. I don't know whether this is relevant, but the integration of limits of each $y_i$ ($i=1,..,k$) varies, that is, some are $(-\infty,\infty)$, others $(0,\infty)$.
From this point, which convergence theorem shall I use to pass the limit inside the integral? I thought of Fatou's Lemma (and reserve Fatou's Lemma) to find the same lower and upper bounds for the integral, but I am sure which conditions (?) I must impose to $\phi(\mathbf{y})$ to apply Fatou's Lemma. Clearly, $\psi$ must be $\phi$-integrable, but is that it?. According to the assumptions, which convergence theorem would be better to apply? In summary, I would to formalize this result. Thanks in advance.