Is it possible to conclude that $a$ is finite and not equal infinity

Question

Assuming that $f_{k}=Ω(g_{k})$ where $f_{k}$, $g_{k}$ are two strictly positive non equal real sequences and $Ω$ is the Big Omega in complexity theory (https://en.wikipedia.org/wiki/Big_O_notation#Family_of_Bachmann%E2%80%93Landau_notations) then $$\liminf_{k \to \infty} f_{k}/g_{k}=a>0$$

Assuming that $\lim_{k \to \infty} f_{k} = 0$

My question is: Is it possible to conclude that $a$ is finite and not equal infinity.

Answer 1

I may miss something. But what about $$\begin{cases} f_k &= 1/k\\ g_k &= 1/k^2 \end{cases}$$