Mdthbs

How is it possible to erase the line (keep it short) that goes up to infinity and use an infinity symbol to indicate so? (Do not touch the rest of the plot and the inserted numbers please if possible.)

documentclass{standalone}

usepackage{expl3}

ExplSyntaxOn

int_zero_new:N g__prg_map_int

ExplSyntaxOff

usepackage{amsmath}

usepackage{amssymb}

usepackage{graphicx}

usepackage{tikz}

usetikzlibrary{arrows,shapes,positioning}

usepackage{pgfplots}

usetikzlibrary{calc}

begin{document}

newcommandddfrac[2]{frac{displaystyle #1}{displaystyle #2}}

pagestyle{empty}

centering

begin{tikzpicture}[yscale=1,xscale=0.05]

 draw[thick,->,>=stealth] (-10,0) -- (45,0);

 draw[thick] (0,-2) -- (0,5);

 draw [smooth,very thick,domain=0.1:40,black] plot (x,{((1*exp(-x/10))/(1-1*exp(-x/10)))+((0.00001*exp(-x/10))/((1-1*exp(-x/10))*(1-1*exp(-x/10))))});

 end{tikzpicture}

 end{document} 

edit: it seems that my solution is almost the same as from @marmoth (+1). In time of uploading mine, I was not aware about this.

I understand that you looking for something like this:

With use of clip function:

documentclass[margin=3mm]{standalone}

usepackage{pgfplots}

usetikzlibrary{arrows,

                calc,

                positioning,

                shapes}

begin{document}

newcommandddfrac[2]{frac{displaystyle #1}{displaystyle #2}}

pagestyle{empty}

centering

begin{tikzpicture}[yscale=1, xscale=0.1]

clip (-1,-1.75) rectangle (50,5);

 draw[-stealth] (-1,0) -- (45,0);

 draw[-stealth] (0,-1.75) -- (0,5);

 draw [domain=0.1:40, smooth,very thick] plot (x,{((1*exp(-x/10))/(1-1*exp(-x/10)))+((0.00001*exp(-x/10))/((1-1*exp(-x/10))*(1-1*exp(-x/10))))});

 draw [domain=0.1:40, samples=51,red] plot (x,{((1*exp(-x/10))/(1-1*exp(-x/10)))+((0.00001*exp(-x/10))/((1-1*exp(-x/10))*(1-1*exp(-x/10))))});

node[below right] at (1,5.1) {$uparrowinfty$};

 end{tikzpicture}

 end{document}

Selection smooth in your draw plot options is wrong. Correct result is when you replace it with selected number of samples. for example with samples=51. See red curve.

I am assuming that you do not want to use pgfplots even though you are loading it. In that case you could clip and intersect.

documentclass[tikz]{standalone}

usetikzlibrary{positioning,intersections}

begin{document}

begin{tikzpicture}[yscale=1,xscale=0.1]

 draw[thick,->,>=stealth] (-10,0) -- (45,0);

 draw[thick] (0,-2) -- (0,5);

 begin{scope}

 clip[name path global=box] (-10,-2) rectangle (45,5);

 draw[smooth,very thick,domain=0.1:40,samples=101,black,name path global=curve] 

 plot (x,{((1*exp(-x/10))/(1-1*exp(-x/10)))+((0.00001*exp(-x/10))/((1-1*exp(-x/10))*(1-1*exp(-x/10))))});

 end{scope}

 draw[-stealth,very thick,name intersections={of=box and curve}] 

 ([yshift=-0.1pt]intersection-1) -- ++(0,0.5)  node[midway,right]{$infty$};

end{tikzpicture}

end{document} 

With pgfplots it would be much easier.

documentclass[tikz]{standalone}

usetikzlibrary{positioning,intersections}

usepackage{pgfplots}

pgfplotsset{compat=1.16}

begin{document}

begin{tikzpicture}

 begin{axis}[axis lines=middle,xtick=empty,ytick=empty,ymin=-2,ymax=5,

    xmin=-2,xmax=45]

  addplot[domain=0.1:45,samples=101,smooth,name path global=curve,

  very thick] {((1*exp(-x/10))/(1-1*exp(-x/10)))+((0.00001*exp(-x/10))/((1-1*exp(-x/10))*(1-1*exp(-x/10))))};

  path[name path global=top] (-2,4.99) -- (45,4.99);

 end{axis}

 draw[-stealth,very thick,name intersections={of=top and curve}] 

 ([yshift=-0.1pt]intersection-1) -- ++(0,0.5)  node[midway,right]{$infty$};

end{tikzpicture}

end{document} 

Interestingly the plots are not the same, most likely because of the better precision of pgfplots. Mathematica seems to be on the side of pgfplots. So I increased the samples in the TikZ picture to match this (but I feel pgfplots is more elegant anyway).