Limit problem with summation: $\lim_{n\to\infty} \frac{1}{n^2 +n} + \frac{2}{n^2 +n} + \dots + \frac{n}{n^2 +n}$ $\lim_{n\to\infty} \frac{1}{n^2 +n} + \frac{2}{n^2 +n} + \frac{3}{n^2 +n} + \frac{4}{n^2 +n} + \dots + \frac{n}{n^2 +n}$
question is when we take limit we can seperate things right ? So we can write $\lim_{n\to\infty} \frac{1}{n^2 + n}$ + $\lim_{n\to\infty} \frac{2}{n^2 + n}$ + .... $\lim_{n\to\infty} \frac{n}{n^2 + n}$
if we take limits one by one we get zeroes. 
we get sum = 0 
but if we do sum first than take limit
$\lim_{n\to\infty} \frac{\frac{n.(n+1)}{2}}{n^2 + n}$ with simplification we get 1/2 
so did my first question wrong ? can't we take limits first than do the sum ?

Two other approaches:
1.$$\sum_{k=1}^n \frac{k}{n^2+n}
=  \frac{\sum_{k=1}^n k}{n^2+n}
= \frac{n(n+1)/2}{n^2+n}
=\frac{1}{2}$$
2.
Let $\Delta x = \frac{1}{n}$ and $x_k=k\Delta x$, then we rewrite as a right Riemann sum:
$$\sum_{k=1}^n \frac{k}{n^2+n}
= \underbrace{\frac{1}{1+1/n}}_{\to\, 1} \cdot\underbrace{\sum_{k=1}^n x_k \Delta x}_{\to\,\int_0^1x\;dx \,=\,\frac12}
\to \frac12
$$
