Welcome! The problem is that in the appendix the equation counter gets reset at the start of a new section. According to this answer one way to go is to load chngcntr and then add
\counterwithout{equation}{section}
\addtocounter{equation}{-1}
after \end{appendix}.
\documentclass[aps,twocolumn,showpacs,amsmath,amssymb,nofootinbib]{revtex4-1}
\usepackage{chngcntr}
\begin{document}
\title{I don't want the equation numbers to change by section after the appendix}
\maketitle
\section{Before appendix}
Here is an equation
\begin{equation}
1+1=2
\end{equation}
And another
\begin{equation}
F=ma
\end{equation}
\section{Just before appendix}
Next...
\begin{equation}
E=mc^2
\end{equation}
\begin{appendix}
\section{An appendix}
La la la...everything is normal here
\begin{equation}
\underset{x\rightarrow0}\lim x=0
\end{equation}
\end{appendix}
\counterwithout{equation}{section}
\addtocounter{equation}{-1}
\section{After appendix}
Here is an equation
\begin{equation}
1+1=2
\end{equation}
And another
\begin{equation}
F=ma
\end{equation}
\section{I wanted (3)!}
Next...
\begin{equation}
E=mc^2
\end{equation}
\end{document}

Of course, you will still have the issue that equation numbers are not unique. So you may perhaps prefer something like this:
\documentclass[aps,twocolumn,showpacs,amsmath,amssymb,nofootinbib]{revtex4-1}
\usepackage{chngcntr}
\newcounter{lastequationbeforeappendix}
\begin{document}
\title{I don't want the equation numbers to change by section after the appendix}
\maketitle
\section{Before appendix}
Here is an equation
\begin{equation}
1+1=2
\end{equation}
And another
\begin{equation}
F=ma
\end{equation}
\section{Just before appendix}
Next...
\begin{equation}
E=mc^2
\end{equation}
\setcounter{lastequationbeforeappendix}{\number\value{equation}}
\begin{appendix}
\section{An appendix}
La la la...everything is normal here
\begin{equation}
\underset{x\rightarrow0}\lim x=0
\end{equation}
\end{appendix}
\counterwithout{equation}{section}
\setcounter{equation}{\number\value{lastequationbeforeappendix}}
\section{After appendix}
Here is an equation
\begin{equation}
1+1=2
\end{equation}
And another
\begin{equation}
F=ma
\end{equation}
\section{I wanted (3)!}
Next...
\begin{equation}
E=mc^2
\end{equation}
\end{document}

\undersetwith\lim. This should work:\lim_{x\to 0}. (Similar to\sum, lower limits are input as subscripts, but set underneath in a display environment.) – barbara beeton Nov 03 '19 at 02:09