Editing Sign function (section)

=== Smooth approximations and limits ===
The signum function can be given as a number of different (pointwise) limits:
<math display="block">\begin{align}
\sgn x &= \lim_{n\to\infty}\frac{1-2^{-nx}}{1+2^{-nx}}\\
       &= \lim_{n\to\infty}\frac{2}{\pi}\operatorname{arctan}(nx)\\
       &= \lim_{n\to\infty}\tanh(nx)\\
       &= \lim_{\varepsilon\to 0} \frac{x}{\sqrt{x^2 + \varepsilon^2}}.
\end{align}</math>
Here, <math>\tanh</math> is the [[hyperbolic tangent]], and <math>\operatorname{arctan}</math> is the [[arctan|inverse tangent]].  The last of these is the derivative of <math>\sqrt{x^2+\varepsilon ^2}</math>. This is inspired from the fact that the above is exactly equal for all nonzero <math>x</math> if <math>\varepsilon=0</math>, and has the advantage of simple generalization to higher-dimensional analogues of the sign function (for example, the partial derivatives of <math>\sqrt{x^2+y^2}</math>).

See ''{{section link|Heaviside step function#Analytic approximations}}''.