equal spacing for 'ordinary' limit and one-sided limit

Question

The problem is that one-sided limits have subscripted symbols above $a$. I tried adding a blank subscript to the 'ordinary' limit, but that didn't seem to even out spacing.

\underline{Theorem:} $\lim \limits_{x \to a^{}} f(x)
\Longleftrightarrow \lim \limits_{x \to a^{-}} f(x)
= \lim \limits_{x \to a^{+}} f(x)$

Answer 1

Use \vphantom{+} to have the same height as the RHS.

\underline{Theorem:} $\lim\limits_{x \to a^{\vphantom{+}}} f(x)
\Longleftrightarrow \lim\limits_{x \to a^{-}} f(x)
= \lim\limits_{x \to a^{+}} f(x)$

Answer 2

Here's a possibility, but takes a lot of work to get there. It uses the \vphantom mentioned by AboAmmar, but it also horizontally centers the \to under the \lim.

It does it by placing a phantom scripted - to the left of the one-sided limits. But then it must also introduce \!\!\! negative space to compensate.

\documentclass{article}
\begin{document}
\underline{Theorem:} $\lim \limits_{x \to a^{\vphantom{-}}} f(x)
\Longleftrightarrow \!\!\!\lim \limits_{^{\phantom{-}}x \to a^{-}} f(x)
= \!\!\!\lim \limits_{^{\phantom{-}}x \to a^{+}} f(x)$
\end{document}

Answer 3

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}