Let $\lim_{L\to\infty}\delta_L=0$ and $\lim_{L\to\infty}L^{d/2}=\infty, d\geqslant 2~~~(*)$
Now two cases are distinguished.
case I
$\lim_{L\to\infty}\delta_L^3L^d=0$
caseII
The $\delta_L$'s are larger such that $\limsup_{L\to\infty}\delta_L^3L^d>0$.
My (maybe silly) question is: If $(*)$ how is it then possible that
(1.) $$\lim_{L\to\infty}\delta_L^3L^d=0$$
resp.
(2.) The $\delta_L$'s are larger such that $\limsup_{L\to\infty}\delta_L^3L^d>0$.
?
And why are this all cases?
$\delta_L=1/L^{d/3}$, then $\delta_L^3L^{d/2}=L^{-d/2}\rightarrow 0$.
$\delta_L=1/L^{d/9}$, then $\delta_L^3L^{d/2}=L^{d/6}\rightarrow\infty$