$\underset{n→∞}{\lim}\frac{1}{n}(1+\sqrt{\frac{n}{n+1}}+\sqrt{\frac{n}{n+2}}+...+\sqrt{\frac{n}{4n-3}})$ is equal to: 

2

$\underset{n→∞}{\lim}\frac{1}{n}[\frac{1}{\sqrt{1+\frac{0}{n}}}+\sqrt{\frac{1}{1+\frac{1}{n}}}+...+\sqrt{\frac{1}{1+\frac{3n-3}{n}}}]=\underset{n→∞}{\lim}\frac{1}{n}\sum\limits_{r=0}^{3n-3}\sqrt{\frac{1}{1+\frac{r}{n}}}=\int\limits_0^3\frac{1}{\sqrt{1+x}}dx=2\sqrt{1+x}|_0^3=2(2-1)=2$