Challenge your limit

$\begin{aligned} L &= \lim_{n \to \infty} \frac 1{n^\frac 32} \sum_{k=1}^n k^\frac 12 \\ &= \lim_{n \to \infty} \frac 1n \sum_{k=1}^n \left(\frac kn\right)^\frac 12 & \small \color{#3D99F6} \text{By Riemann's sums:} \\ &= \int_0^1 x^\frac 12 dx & \small \color{#3D99F6} \lim_{n \to \infty} \frac 1n \sum_{k=a}^b f \left(\frac kn\right) = \lim_{n \to \infty} \int_\frac an^\frac bn f(x) \ dx \\ &= \frac 23 x^\frac 32 \bigg|_0^1 \\ & = \frac 23 \approx \boxed{0.667} \end{aligned}$