Evaluating limit by sandwich theorem: $\lim_{n\to\ \infty}\frac{1}{1+n^2} +\frac{2}{2+n^2}+\cdots+\frac{n}{n+n^2}$ $$\lim_{n\to\ \infty}\frac{1}{1+n^2} +\frac{2}{2+n^2}+\cdots+\frac{n}{n+n^2}$$
For using Sandwich theorem I need two functions such that $g(x)<f(x)<h(x)$
$$\frac{1}{n+n^2} +\frac{2}{n+n^2}+\cdots+\frac{n}{n+n^2} \leq \frac{1}{1+n^2} +\frac{2}{2+n^2}+\cdots+\frac{n}{n+n^2} $$
But I can't find function greater than the given which will help me evaluate the limit.

Your expression may be written as $$\lim_{n\to \infty}\sum_{k=1}^{n}\frac{k}{k+n^2}$$ so, you can search for terms $a(k,n)$ and $b(k,n)$ such that $$a(k,n)\le \frac{k}{k+n^2}\le b(k,n)$$ For example $$a(k,n)=\frac{k}{n+n^2} \text{ or } a(k,n)=\frac{k}{2n^2}$$ and $$b(k,n)=\frac{k}{k+k^2} \text{ or } b(k,n)=\frac{k}{n^2}$$
