Plot of 1/x using pgfplots

I noticed I couldn't find pgfplots version of the $1/x$ plot that describes $e$. I needed one, so here is the LaTeX I used.

\documentclass[margin=3mm]{standalone}

\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepgfplotslibrary{fillbetween}

\begin{document


\begin{tikzpicture}
\begin{axis}[
ytick = {1,2},
xtick = {0,1,2,3},
xlabel=$x$, ylabel=$y$,
ymax=2.5, xmin=0, ymin=0,
xlabel style = {anchor=north east},
ylabel style = {anchor=north east}
]
\addplot [color = blue, name path=A,domain=0:4, line width = 1.4pt, samples=200] {1/x};
\path [name path=B]
    (\pgfkeysvalueof{/pgfplots/xmin},0) --
    (\pgfkeysvalueof{/pgfplots/xmax},0);
\addplot [blue!20]
    fill between [of=A and B,
                  soft clip={(1,0) rectangle (2.71,25)},];
\node at (axis cs:1.5,1.2) {$y=1/x$};
\node at (axis cs:1.6,0.31) {Area = $1$};
\node at (axis cs:2.71, 0.82) {$e$};
\addplot[-latex] coordinates
           {(2.71,0.75) (2.71,0.45)};
\end{axis}
\end{tikzpicture}

\end{document}