Restricting tikz plot to multiple domains due to dividing by zero

Question

I'm trying to plot a formula where for my intended domain, a division by zero will be done. Overleaf/latex gives me the following error: Dimension Too Large. I think this has to do with each other. That is why I try restrict the plotting around the asymptotes. The mathematical problem is that there are poles at +3.5 and -3.5 ; here is a link to wolframalpha, showcasing the function: link

Inspired by link, I tried to implement it the following way, without success.

Do you have suggestions how I can restrict an tikz plot for multiple domains. I could split the function into multiple \addplot. However, I do not prefer this, as in my real report I'm plotting a lot off functions

\begin{figure}[ht]
    \begin{tikzpicture}
        \begin{axis}[
            xmin=-9, xmax=9,
            ymin=0, ymax=5,
            % domain = -9:9
            ]
            
            \addplot [%restrict expr to domain={<expr>}{min:max}
                    %   restrict expr to domain={(x>=-9)&&(x<=-3.4)||(x>=-3.6)&&(x<=3.4)||(x>=3.6)&&(x<=9)}{1:1},
                      samples=100, 
                      line width=0.45mm, 
                      color=blue,
            ]
            {(3)/(x+3.5)^2 + (3)/(3.5-x)^2 + 3 * e^(-((0-x)^2)/ (2 * 1^2))};
           
       
        \end{axis}
    \end{tikzpicture}
\end{figure}

Answer 1

Like this?

By use of restrict y to domain =<min>:<max> ,:

\documentclass[border=3.141592]{standalone}
\usepackage{pgfplots}                             
\pgfplotsset{compat=1.18}

\begin{document}
    \begin{tikzpicture}
\begin{axis}[
    axis lines=center,
    xlabel = {$x$}, xlabel style = {anchor=west},
    ylabel = {$y$},
    xmin=-9, xmax=9, xtick={-8,-6,...,8},
    extra  x ticks = {0},
    ymin=0,  ymax=27,
    tick label style = {font=\scriptsize},
    domain = -8.5:8.5,
    restrict y to domain = 0:100,
    no marks,
every axis plot post/.append style={very thick},
            ]
\addplot +[samples=101]
            {(3)/(x+3.5)^2 + (3)/(3.5-x)^2 + 3*e^(-((0-x)^2)/(2 * 1^2))};
\end{axis}
    \end{tikzpicture}
\end{document}