Evaluate the following limit by first recognizing the sum as a Riemann sum for a function defined on $[0,1]$ :
\[
\lim _{n \rightarrow \infty} \frac{1}{n}\left(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\sqrt{\frac{3}{n}}+\cdots+\sqrt{\frac{n}{n}}\right)
\]

Step 1 ：Recognize the given limit as a Riemann sum for the function \(f(x) = \sqrt{x}\) on the interval \([0,1]\).

Step 2 ：The sum inside the parentheses is the sum of function values at points in the interval, and the factor of \(\frac{1}{n}\) outside the parentheses corresponds to the width of the subintervals of the partition.

Step 3 ：The limit of this Riemann sum as \(n\) approaches infinity is equal to the definite integral of \(f(x)\) from \(0\) to \(1\).

Step 4 ：Calculate the integral of \(\sqrt{x}\) from \(0\) to \(1\).

Step 5 ：The integral of \(\sqrt{x}\) from \(0\) to \(1\) is \(\frac{2}{3}\).

Step 6 ：Therefore, the limit of the given Riemann sum as \(n\) approaches infinity is also \(\frac{2}{3}\).

Step 7 ：Final Answer: \(\boxed{\frac{2}{3}}\)