Comparing efficiency between estimators

Suppose that $\hat \theta_1, \hat \theta_2$ are two estimators of $\theta$ . Furthermore, assume that \begin{align} \sqrt{n}(\hat \theta_1-\theta)\overset{d}{\to}N(0,V_1)\\ \sqrt{n}(\hat \theta_2-\theta-B)\overset{d}{\to}N(0,V_2), \end{align} where $V_{1}=\lim_{n\to\infty} V_{1,n}$ , $V_{2}=\lim_{n\to\infty} V_{2,n}$ and $B:=B_n$ are all known (including $V_{1,n},V_{2,n}$ ). Does it make sense to talk about relative efficiency between $\hat \theta_1$ and $\hat \theta_2$ by comparing $V_1$ and $V_2$ (or even $V_{1,n}$ and $V_{2,n}$ ) in the presence of term $B$ ? For example, if I show that $V_{1,n}/n-V_{2,n}/n-B , then can I say that $\hat\theta_1$ is relatively more efficient that $\hat\theta_2$ ? *Based on the answer and a comment, I clarify that the term $B$ is relevant in the expression for $\hat \theta_2$ . Hence it depends on $n$ . $B$ itself is $o(1), n\to\infty$ .