Of course it is possible, but much caution is required because we don't know about uniformity in $k$ for the estimates corresponding to the $\sim$ statements. Counterexamples are very easy to construct.

Of course it is possible, but much caution is required because we don't know about uniformity in $k$ for the estimates corresponding to the $\sim$ statements. Counterexamples are very easy to construct, where e.g. $f_k(x)$ doesn't start behaving "nicely" until $x > k$.