Chapter 5, Problem 30RE

In each part, find the limit by interpreting it as a limit of Riemann sums in which the interval $\left[0,1\right]$ is divided into $n$ subintervals of equal length.

(a) $\underset{n\to +\infty }{\lim }\frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\cdots +\sqrt{n}}{n^{3/2}}$

(b) $\underset{n\to +\infty }{\lim }\frac{1^4+2^4+3^4+\cdots +n^4}{n^5}$

(c) $\underset{n\to +\infty }{\lim }\frac{e^{1/n}+e^{2/n}+e^{3/n}+\cdots +e^{n/n}}{n}$