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: 
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:
$${\pi\over2}-\arctan x=O\left({1\over x}\right),\quad 1-\tanh x=O(e^{-x})\qquad(x\to \infty)\ .$$
Reflecting about the line $x=y$, $y=\tan(x)$ becomes $y=\arctan(x)$:

![enter image description here](https://i.stack.imgur.com/wmQBz.png)

However, to get something of the shape in the given link, we can use
$$
f(x)=\frac{x^2}{1-2x+2x^2}
$$
![enter image description here](https://i.stack.imgur.com/BqCSv.png)