Calculating $\lim\limits_{j \to \infty} \frac{j^{j/2}}{j!}$

Question

\begin{align*}
& \lim\limits_{j \to \infty}{j^{\,j/2} \over j!} \\
\end{align*}

This problem is from a real analysis textbook in the chapter on the natural log and properties of exponents. I'm struggling with how to approach this. I don't think you would use L'Hopital's rule.

Answer 1

The ratio test is a more elementary tool: $$\frac{(j+1)^\tfrac{j+1}2}{(j+1)!}\,\frac{j!}{j^{\tfrac j2}}=\frac{(j+1)^{\tfrac{j}2}\sqrt{j+1}}{(j+1)j^{\tfrac j2}}=\underbrace{\sqrt{\Bigl(1+\frac1j\Bigr)^j}}_{\begin{array}{c}\downarrow\\\sqrt{\mathrm e}\end{array}}\,\frac1{\sqrt{j+1}}\to 0$$