Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$ Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to
\infty} s_n$. 
 \begin{align}
  s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx
 \end{align}
I have written a solution of my own, but I would like to know it is completely correct, and also I would like if people post more alternative solutions. 

Using the substitution $x\mapsto x^{1/n}$ and Dominated Convergence,
$$
\begin{align}
\lim_{n\to\infty}\int_0^1\frac{nx^{n-1}}{1+x}\,\mathrm{d}x
&=\lim_{n\to\infty}\int_0^1\frac1{1+x^{1/n}}\,\mathrm{d}x\\
&=\int_0^1\frac12\,\mathrm{d}x\\
&=\frac12
\end{align}
$$
