$\newcommand\R{\Bbb R}$For $a=(a_1,\dots,a_n)\in\R^n$, let $$L(a):=\lim_{x\downarrow0}R_a(x)$$ if the limit exists, where $$R_a(x):=\frac{x^a}{F(x)},$$ $x=(x_1,\dots,x_n)\in\R^n$, $x\downarrow0$ means that $x_i\downarrow0$ for each $i\in[n]:=\{1,\dots,n\}$, $x^a:=x_1^{a_1}\dotsm x_n^{a_n}$, $F(x):=\sum_{k\in S}F_k x^k$, $S$ is a finite nonempty subset of $\R^n$, and $F_k\in(0,\infty)$ for all $k=(k_1,\dots,k_n)\in S$.
Is it then always true that $L(a)=0$ iff $$l_p(a):=\lim_{t\downarrow0}R_a((t^{p_1},\dots,t^{p_n}))=0\text{ for all $p=(p_1,\dots,p_n)\in(0,\infty)^n$?}$$
This question is motivated by this previous one.
