Evaluate:

$\underset{n\to \mathrm{\infty}}{lim}{\int}_{0}^{\mathrm{\infty}}{(1+\frac{x}{n})}^{-n}\mathrm{sin}\frac{x}{n}\phantom{\rule{thinmathspace}{0ex}}dx.$

I have tried to use the (Lebesgue) dominated convergence theorem to evaluate the same. At first I noticed that:

$\begin{array}{}& |{(1+\frac{x}{n})}^{-n}\mathrm{sin}\frac{x}{n}|\le \left|{(1+\frac{x}{n})}^{-n}\right|\le 1\text{for all positive}x\text{and for all natural}n.\end{array}$

Since $g(x)=1$ is Lebesgue integrable for each $x\in [0,\mathrm{\infty}).$ So by DCT, the given limit equals:

${\int}_{0}^{\mathrm{\infty}}\underset{n\to \mathrm{\infty}}{lim}{(1+\frac{x}{n})}^{-n}\mathrm{sin}\frac{x}{n}\phantom{\rule{thinmathspace}{0ex}}dx={\int}_{0}^{\mathrm{\infty}}{e}^{-x}\cdot 0\phantom{\rule{thinmathspace}{0ex}}dx=0.$

This question was asked in our end term exam this semester and so I don't know the correct answer to this question. Therefore, just to check whether I have evaluated the limit correctly I am posting the same here on MSE. If I have gone wrong somewhere, please point out and give some insights. Thanks in advance.

$\underset{n\to \mathrm{\infty}}{lim}{\int}_{0}^{\mathrm{\infty}}{(1+\frac{x}{n})}^{-n}\mathrm{sin}\frac{x}{n}\phantom{\rule{thinmathspace}{0ex}}dx.$

I have tried to use the (Lebesgue) dominated convergence theorem to evaluate the same. At first I noticed that:

$\begin{array}{}& |{(1+\frac{x}{n})}^{-n}\mathrm{sin}\frac{x}{n}|\le \left|{(1+\frac{x}{n})}^{-n}\right|\le 1\text{for all positive}x\text{and for all natural}n.\end{array}$

Since $g(x)=1$ is Lebesgue integrable for each $x\in [0,\mathrm{\infty}).$ So by DCT, the given limit equals:

${\int}_{0}^{\mathrm{\infty}}\underset{n\to \mathrm{\infty}}{lim}{(1+\frac{x}{n})}^{-n}\mathrm{sin}\frac{x}{n}\phantom{\rule{thinmathspace}{0ex}}dx={\int}_{0}^{\mathrm{\infty}}{e}^{-x}\cdot 0\phantom{\rule{thinmathspace}{0ex}}dx=0.$

This question was asked in our end term exam this semester and so I don't know the correct answer to this question. Therefore, just to check whether I have evaluated the limit correctly I am posting the same here on MSE. If I have gone wrong somewhere, please point out and give some insights. Thanks in advance.