Calculus limit with sum of radicals

Question

I am trying to calculate the following limit, but to be honest I've been failing miserably.

$\lim\limits_{n\rightarrow\infty}\frac{\sqrt{1+\frac{1}{n}}+\sqrt{1+\frac{2}{n}}+...+\sqrt{1+\frac{n}{n}}}{n}$

I've been trying to use the fact below but got me no useful result:

$\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$

Also I've been trying to use the logarithm to obtain a product but still no results.

Any hints, please?

Update 1:

I followed your advice and identified the function $f(x)=\sqrt{1+x}, f:[0,1]$

So I guess now the result of that limit is $\int_0^1{\sqrt{1+x}}$ ?

Answer 1

Hint:

Rewrite it as $$\frac1n\sum_{k=1}^n\sqrt{1+\frac kn}$$ and you should see a Riemann sum of a certain function on a certain interval.