What function looks like a tangent on its side?

Question

Basically, I want a curve that decreases toward a lower limit as x approaches negative infinity and rises toward an upper limit as x approaches positive infinity - It would have this general shape:

https://en.wikipedia.org/wiki/Contract_curve

The closest I could find to a description is a non-repeating tangent laid on its side.

I am sure I have seen something similar before; I just wish I had gone further in my study of trig and calculus...

Answer 1

Two examples are $$f(x):=\arctan x\qquad(-\infty<x<\infty)$$ with $\lim_{x\to\pm\infty}f(x)=\pm{\pi\over2}$, and $$g(x):=\tanh x\qquad(-\infty<x<\infty)$$ with $\lim_{x\to\pm\infty}g(x)=\pm1$. Their graphs look similar, but there is an essential difference: $${\pi\over2}-\arctan x=O\left({1\over x}\right),\quad 1-\tanh x=O(e^{-x})\qquad(x\to \infty)\ .$$

Answer 2

For example $\sqrt{x^2+1}/x$ will be between -1 and 1.

Answer 3

Reflecting about the line $x=y$, $y=\tan(x)$ becomes $y=\arctan(x)$:

However, to get something of the shape in the given link, we can use $$ f(x)=\frac{x^2}{1-2x+2x^2} $$