A Problem on Limit from an old JEE

$\lim_{n\to\infty} \cfrac{1}{n} \sum_{r=1}^{2n}\cfrac{r}{\sqrt{r^{2}+n^{2}} } =lim_{n\to\infty} \cfrac{1}{n} \sum_{r=1}^{2n}\cfrac{\cfrac{r} {n}}{\sqrt{\cfrac{r^{2}} {n^{2}}+1}}$ According to Riemann's integral, $lim_{n\to\infty}\cfrac{1}{n}\sum_{r=1}^{2n}\cfrac{\cfrac{r}{n}}{\sqrt{\cfrac{r^{2}}{n^{2}}+1}}=\int_{0}^{2} \cfrac{x} {\sqrt{1+x^{2}}} dx=\cfrac{1} {2} \int_{0}^{2} \cfrac{2x} {\sqrt{1+x^{2}}}dx$

Let $y=x^{2}$

$\cfrac{1} {2} \int_{0}^{2} \cfrac{2x} {\sqrt{1+x^{2}}} dx=\cfrac{1} {2} \int_{0}^{4} \cfrac{1} {\sqrt{1 +y}}dy =[\sqrt{1+y}]_{0}^{4}=\sqrt{5} - 1$ Hence, the required value of the original limit is $\sqrt{5}-1$ .