Given a function $f \in L^1(\mathbb{R})$, suppose I want to evaluate the following integral
$$\lim_{n\to\infty} \int_{\mathbb{R}}e^{-nx^2}f(x)dx$$
It looks to me like one could use dominated convergence here. If we let $f_n(x) = e^{-nx^2}f(x)$, then clearly $|f_n(x)| \leq |f(x)|$ for $n \geq 0$, and we know that $|f(x)|$ is an integrable function. The issue I am having is showing that $f_n(x)$ itself is integrable - it seems clear to me that it would be, but not sure how to show it (maybe it just follows from the above inequality since $|f(x)|$ is integrable?)
Also, suppose we can show dominated convergence theorem applies, the limit is a little bit interesting as it depends on $x$. Clearly, if $x\neq 0$, then we have
$$\lim_{n\to\infty} e^{-nx^2}f(x) = 0,$$ but if $x = 0$, then what happens?
Thanks!
$f_n$ is integrable because $|f_n| \le |f|$, as you mentioned.
We have $$\lim_{n \to \infty} f_n(x) = \begin{cases} 0 & x \ne 0 \\ f(0) & x = 0. \end{cases}$$
The integral of this function is simply zero. So if you show that you can use the dominated convergence theorem, then $\lim_{n \to \infty} \int f_n = 0$.