Assume that we have a monotonously increasing sequence $L_k$, such that $\lim_{k\rightarrow \infty}\left(L_kb^{-k}\right)=0$ and $\lim_{k\rightarrow \infty}\left(L_ka^{-k}\right)=\infty$, can we conclude that:
a) there exists $a\leq c\leq b$, such that $\lim_{k\rightarrow \infty}\left(L_kc^{-k}\right)=0$ and $\lim_{k\rightarrow \infty}\left(L_k(c-\varepsilon)^{-k}\right)=\infty$, and
b) that $L_k=Cc^k+o(c^k)$, where C is some constant?
Thank you very much in advance for any help. (This is a follow up to my earlier question here: asymptotics of a sequence)
Consider $\ln n$, $a=1$, $b=2$. There is no $c$.