It is proved that if $f\in K'_1(R)$ $(f\in S'(R))$ has the $S$-asymptotic in $D'(R)$ related to a function $c(h)\in\Sigma_e(R)$ $(c(h)\in\Sigma_e(R))$, $h\to\infty$, then f has the $S$-asymptotic in $K'_1(R)$ (in $S'(R)$) related to $c(h)$. For the same assertion in the $n$-dimensional case $n>1$, some additional assumptions are needed.