If a
is a constant, I'd suggest you use \uparrow
and \downarrow
to indicate unambiguously that the x
is supposed to approach a
from below and above, respectively. A very nice side-effect of this notation is that x\uparrow a
and x\downarrow a
take up less space than x\to a
-- obviating any need to make spacing adjustments.
If, in addition, you wish to assure that x\to a
and x\uparrow a
(and x\downarrow a
) are typeset at same distance below "lim", just change x\to a
to x\to a\vphantom{\uparrow}
.

\documentclass{article}
\begin{document}
$\displaystyle
\lim_{x\to a} f(x)\Longleftrightarrow
\lim_{x\uparrow a} f(x)=
\lim_{x\downarrow a} f(x)$
\end{document}
\limits_{x \to a^{\phantom{-}}}
to leave extra space in the ordinary limit, but I don't like the look of it at all. – Steven B. Segletes Dec 18 '18 at 15:31