It does seem like a very simple question, but I don't see how to do it properly.
How can we show that the limit of
$$3^{\lfloor\frac{-\log(j)}{\log(2)}\rfloor}j^{\frac{\log(3)}{\log(2)}}$$
as $j\rightarrow\infty$does not exist, while $\limsup$ and $\liminf$ exist and are equal to $1$ and $\frac{1}{3}$?
I can see it intuitively but don't see how to prove it. Thank you very much for any help.
Every term of the sequence is in $(\frac13,1]$ since $x-1\lt\lfloor x\rfloor\leqslant x$ for every $x$. If $j=2^n$, the result is $1$. If $j=2^n+1$, the result is $\frac13\,\left(1+\frac1{2^n}\right)$. Hence the limsup is $1$ and the liminf is $\frac13$.