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.

The reasoning is not correct because $$ \int_0^{+\infty} \frac{1}{1+x} = + \infty $$
And thus $\frac{1}{1+x}$ is not integrable.
Hint for a correct reasoning:
For $n \geq 2$ we have $(1+y)^n \geq 1 + n \cdot y + \frac{n \cdot (n-1)}{2} \cdot y^2 \space \space \space \forall y \geq 0$
