I recently saw a post in which the query was about a function that tends to the Dirac delta function as a parameter in it tends to infinity. The function chosen was $${(1+\cos x)^n\over C}$$ as $n\to\infty$, with C being the integral of the given function from $-\pi$ to $+\pi$. The specific query of the post was infact to do with this normalisation constant. Another way of doing this would be with a parametrised Gaussian $$\sqrt\frac{\alpha}{\pi}\exp(-\alpha x^2)$$ as $\alpha\to\infty$.
Various other non smooth functions can be represented as limits of parametrised smooth functions as the parameter tends to infinity: |x| can be represented as $${1\over\alpha}\ln\cosh(\alpha x)$$ (or as $${1\over\alpha}\ln(2\cosh(\alpha x))$$ ... it doesn't really matter whether the 2 is there or not; the sharp-cornered-sigmoid function ( $$x\leq -1 ⇒y=-1 ,$$$$ -1<x<+1⇒y=x, \&$$$$x\geq 1⇒y=+1$$ by $$\frac{1}{\alpha}\left(\ln\left(1+\exp(\alpha(1+x))\right)-\ln\left(1+\exp(\alpha(1-x))\right)\right)-x .$$ or, equivalently
$$\frac{1}{\alpha}\ln\frac{1+\exp(\alpha(1+x))}{1+\exp(\alpha(1-x))}-x .$$
Does anyone know any other limits of parametrised smooth functions for representing either these or other non-smooth functions in this kind of way?
I have found these kinds of function handy in computer graphics; and also in the numerical solution of differential equation: if, say, a forcing function or boundary condition is a non-smooth function of which the being encoded raw into the solution algorithm would cause the propagation of artifacts, then a function such as one of these could be coded-in instead ... and it could even be fine-tuned such that the sharpnesses of it be as much as the algorithm can tolerate.
And plugging the one for $|x|$ into the one for the sharp cornered sigmoid, and turning the result upside-down, you could get a unit tent function with $$1-{1\over\alpha}\ln\frac{1+e^\alpha \cosh(\alpha x)}{e^\alpha+\cosh(\alpha x)} ,$$ which you might as well reduce to $$1-{1\over\alpha}\ln\frac{e^\alpha \cosh(\alpha x)}{e^\alpha+\cosh(\alpha x)} $$ or $$1+{1\over\alpha}\ln(e^{-\alpha}+\operatorname{sech}(\alpha x)) .$$
To get the heavyside step function, any unboundedly 'encelerated' sigmoid function can be used, with a little scaling & vertical displacement, such as $$\frac{1}{2}+\frac{1}{\pi}\operatorname{atn}(\alpha x)$$ or $$\frac{1}{2}(1+\operatorname{tanh}(\alpha x)) .$$ The function $\tanh$ has the theoretical advantage of being one that cleaves more closely to the limits $y=-1$ & $y=+1$; but on the other hand, it's worth keeping both of these in mind, as in a use of such as these in practice for the kind of thing I have given a use of them for, the greater 'slackness' of $\operatorname{atn}$ could actually be an advantage (in fact I have found that sometimes it is). Also in similar fashion a square pulse can be represented by $$\frac{1}{\pi}(\operatorname{atn}(\alpha (1-x))+\operatorname{atn}(\alpha (1+x)))$$ or by $$\frac{1}{2}(\operatorname{tanh}(\alpha (1-x))+\operatorname{tanh}(\alpha (1+x))) .$$