Equation for a modified sigma curve that depicts $y=x$.

Question

I have graphed what I am trying to achieve

I can't embed photos due to not having enough points here, but here is an image of the graph

I have been trying to create a sigma function that maintains y=x until it gets to the limit at which point is smoothly eases to the limit.

I am not using an existing sigma function because they do not maintain $y=x$ and the transition to the limit is large, and sigma functions do not reach the limit while I would like it to reach the limit. It is entirely possible this already exists but I've been searching and can't locate it.

The corner does not necessarily need to be a perfect circle.

Answer 1

This is what I was after, with a high value for $k$ and height as $c$

$y=\frac{x}{\left(1+\left|\frac{x}{c}\right|^{k}\right)^{\frac{1}{k}}}$

https://www.desmos.com/calculator/bige29p1sy