Finding $\lim\limits_{n \to \infty} \int_{-n}^n f(1+\frac{x}{n^2})g(x)\,\mathrm dx$

Question

Given $g:\mathbb{R} \rightarrow \mathbb{R}$ is integrable and $f：\mathbb{R} \rightarrow \mathbb{R}$ is bounded, measurable, and continuous at $1$. Find $$\lim_{n \to \infty} \int_{-n}^n f\left(1+\frac{x}{n^2}\right)g(x)\,\mathrm{d}x.$$

I was thinking of using dominated convergence, but I can't find a dominator…

Answer 1

We have $\int_{-n}^n f\left(1+\frac{x}{n^2}\right)g(x)\,\mathrm{d}x= \int_{\mathbb R} 1_{(-n,n)} f\left(1+\frac{x}{n^2}\right)g(x)\,\mathrm{d}x$.

Now let $f_n(x):=1_{(-n,n)} f\left(1+\frac{x}{n^2}\right)g(x)$.

If $x \in \mathbb R$, then there is $N$ such that $x \in (-n,n)$ for all $n>N$, hence

$f_n(x) = f\left(1+\frac{x}{n^2}\right)g(x)$ for all $n>N$,thus

$f_n(x) \to f(1)g(x)$.

Can you proceed ?

Answer 2

Let $M$ be a bound of $f$ and consider $M\,|g|$.