Let $c_{j}$ be a sequence of positive integers with $c_{j}\leq j$ such that $\lim_{j\rightarrow\infty}\frac{c_{j}}{j}=r\in\left[0,1\right]$, and let $a,b$ be positive reals, with $a<b$.
Is: $$\lim_{j\rightarrow\infty}\left(\frac{a^{c_{j}/j}}{b}\right)^{j}\overset{?}{=}\lim_{j\rightarrow\infty}\left(\frac{a^{r}}{b}\right)^{j}$$
Thanks!
Hint:
Notice that $$ \left( \frac{a^{\frac{c_j}{j}}}{b}\right)^j = \Big( a^{\frac{c_j}{j}- r} \Big)^j \, \Big( \frac{a^r}{b}\Big)^j $$ So, for any case where the second factor converges, you'd need the first factor to converge to $1$. I bet you can find some counter examples, particularly if you look for cases such as when $\frac{c_j}{j} - r \sim j^{-1/2}$ or similar.