Integral as a limit of sum: $\lim\limits_{n\to\infty}\sum_{k=1}^n\left(\frac k{n^2+n+2k}\right)$ 
The value of$$\lim_{n\to\infty}\sum_{k=1}^n\left(\frac k{n^2+n+2k}\right)=\,?$$

Can the limit be partially applied to the denominator after converting the numerator into an integral?
I wrote this as $$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\left(\frac {k/n}{1+1/n+2k/n^2}\right)$$
From what I know I can write that $1/n$ as $\mathrm{d}x$ and $k/n$ as $x$. Can I apply limit to denominator as $n$ tends to infinity and rewrite the denominator as $1$.

What you can also do is to use generalied harmonic numbers since
$$S_n=\sum_{k=1}^n\left(\frac k{n^2+n+2k}\right)=\frac{1}{4} n \left((n+1) H_{\frac{1}{2} n (n+1)}-(n+1) H_{\frac{1}{2} n (n+3)}+2\right)$$
Now, using twice their asymptotics and continuing with Taylor series
$$S_n=\frac{1}{2}-\frac{2}{3 n}+\frac{4}{3
   n^2}+O\left(\frac{1}{n^3}\right)$$
