Btukfyl

I have the following table almost finished (I tried to make the code as much readable as I could)

documentclass{article}

usepackage[table]{xcolor}

usepackage{tikz}

usepackage{mathtools}

usepackage{array}



begin{document}

    begin{center}

        renewcommand{arraystretch}{1.6}

        begin{equation*}

        begin{array}{ | >{centeringarraybackslash$} p{4.0cm} <{$} 

            | >{centeringarraybackslash$} p{4.0cm} <{$} 

            | >{centeringarraybackslash$} p{4.0cm} <{$}

            | } hline

        a+infty=infty & a-infty=-infty & infty+infty=infty \ hline                              % End of 1st row



        -infty-infty=-infty & inftycdotinfty=infty & -inftycdotinfty=-infty \ hline         % End of 2nd row



        dfrac{a}{infty}=0         & 



        dfrac{a}{0}=

        left{ renewcommand{arraystretch}{1}

        begin{tabular}{c}

        $ infty $ si $ a>0 $\

        $ -infty $ si $ a<0 $

        end{tabular} right. &



        infty^a=

        left{ renewcommand{arraystretch}{1}

        begin{tabular}{c}

        $ infty $ si $ a>0 $\

        $ 0 $  si $ a<0 $

        end{tabular}right. \ hline                                                                  % End of 3rd row



        infty^infty=infty&infty^{-infty}=0&



        a^infty=

        left{ renewcommand{arraystretch}{1}

        begin{tabular}{c}

        $ infty $ si $ a>1 $\

        $ 0 $  si $ 0leq a<1 $

        end{tabular}right. \ hline                                                                  % End of 4th row



        cellcolor{gray!25}& 0^a=0text{ si $ a>0 $} &cellcolor{gray!25}  \ hline                       % End of 5th row



        end{array}

        end{equation*}

    end{center}

end{document}

Several problems here:

• I managed to make the columns to have the same width but I have to copy-paste that code every time I want to make a column with that property (maybe some redefinition with renewcommand?)


• When I use tabular inside the array environment the curly brackets are too close from the cell limits, and I wonder if you can redefine the size of all the cells of those rows to fit those tables.


• Is there an option that array (or tabular) could get that stretch the array without having to use renewcommand{arraystretch}{<number>} every time I make a new table?

I'm looking for a general solution, not only for this problem, since I will define a lot of tables in my next report, and for sure a lot of people will appreciate a general solution since I see lots of questions about this mysterious world of LaTeX tables.