Finding a dominating function for this sequence Let $$f_n (x) = \frac{nx^{1/n}}{ne^x + \sin(nx)}.$$
The question is: with the dominated convergence theorem find the limit
$$ \lim_{n\to\infty} \int_0^\infty f_n (x) dx. $$
So I need to find an integrable function $g$ such that $|f_n| \leq g$ for all $n\in \mathbf N$. I tried
$$ \frac{nx^{1/n}}{ne^x + \sin(nx)}
= \frac{x^{1/n}}{e^x + \sin(nx)/n}
\leq \frac{x^{1/n}}{e^x - 1}
\leq \frac{x^{1/n}}{x}.
$$
But I can't get rid of that $n$. Can anyone give me a hint?

We have 
\begin{align}
\left|
\frac{nx^{1/n}}{ne^x + \sin(nx)}
\right|=
&
\frac{|x^{1/n}|}{|e^x + \sin(nx)/n|}
&
\\
\leq 
&
\frac{\max\{1,x\}}{|e^x + \sin(nx)/n|}
&
\mbox{by } |x^{1/n}|\leq \max\{1,x\} 
\\
\leq
& 
\frac{\max\{1,x\}}{|e^x -\epsilon |}
&
\mbox{if } |e^x + \sin(nx)/n|\geq |e^x -\epsilon|
\\
\end{align}
Note that for all $\epsilon\in(0,1)$ existis  $N$ any sufficiently large such that 
$$
\left|\frac{1}{n}\sin(nx)\right|<\epsilon \implies |e^x + \sin(nx)/n|\geq |e^x -\epsilon|
$$
if $n>N$.
