Lebesgue Dominated Convergence Application I want to compute the integral
$$\lim_{n\to \infty}\int_0^\infty \frac{\sin\left(\frac{x}{n}\right)}{(1+x/n)^n}\,\mathrm{d}x$$
Since 
$$\left| \frac{\sin\left(\frac{x}{n}\right)}{\left(1+\frac{x}{n}\right)^n}\right | \le \frac{1}{\left|\left(1+\frac{x}{n}\right)^n\right|}\le \frac{1}{1+x}$$
for $x\in [0,\infty)$, where I used $(1+x)^n \ge 1+xn$ to obtain the last inequality.
I have therefore found a lebesgue integrable upper bound of the series of the integrand. Since the integrand converges to zero, I would obtain the integral to be zero.
Is my reasoning correct?
EDIT: Corrected the inequality.

A lower bound:
$$ I_n = \int_{0}^{+\infty}\frac{\sin\left(\frac{x}{n}\right)}{\left(1+\frac{x}{n}\right)^n}\,dx \geq \int_{0}^{+\infty}\sin\left(\frac{x}{n}\right)e^{-x}\,dx = \frac{n}{n^2+1} $$
and a compatible upper bound:
$$ I_n = n\int_{0}^{+\infty}\frac{\sin(x)}{(1+x)^n}\,dx \leq n\int_{0}^{+\infty}\frac{x}{(x+1)^n}\,dx = \frac{n}{(n-1)(n-2)} $$
prove that the wanted limit is zero and $I_n = \frac{1}{n}\left(1+o(1)\right)$.
