Plot of the hyperbolic tangent. tanh(x) goes to +/- 1 for x to +- infinity while it is approximately linear for abs(x) < 1. Appears in many places in physics, e.g. in the magnetization of an ideal paramagnet of independent spins.
\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
xlabel = $x$,
ylabel = $\tanh(x)$,
ymin = -1.25,ymax = 1.25,
domain = -2:2,
smooth,thick,
axis lines = center,
every tick/.style = {thick}]
\addplot[color=blue]{tanh(x)};
\addplot[color=blue,dashed,domain=-1:1,thin]{x};
\end{axis}
\end{tikzpicture}
\end{document}