Is there a (preferably simple) function that becomes more and more erratic as we take the variable to some limit? So suppose the limit is zero. Each time the function is evaluated with a smaller variable going closer to zero the result will appear to have less and less of a pattern than when evaluated away from that limit. I guess another way to say it is a function that becomes more orderly away from that limit. My initial interest is that I want to use the function as an analogy in an essay, but now I am genuinely interested. When I say preferably simple, I mean not a huge messy equation and also something whose behavior toward the limit does not have infinitesimal behavior such as $\sin(\cot(x))$ toward zero. Below is a messy example of the "erratic" behavior I'm looking for.
Perhaps not a function in the sense you mean, but it may suit your purposes. Consider the borders of (for example) the Mandelbrot Set.
$$z\rightarrow z^2 + C$$
The solution is quite predictable within the centre of the structure where $|z|<<1$ and in the outer reaches, where $|z|>>1$. However, it behaves unpredictably where $|z|\approx 1$.