Wikipedia:Reference desk/Archives/Mathematics/2016 March 23

= March 23 =

Simple monotonic functions that asymptomatically approach a value from above
I'm looking for simple smooth monotonically decreasing functions f(x) that have all the following properties:


 * f(1) = 1
 * as x approaches infinity, f(x) asymptomatically approaches, from above, a constant c (obviously, the previous condition implies c < 1; I'd be interested both in functions that only work for c >= 0 and in functions that work even for negative c)

What are the simplest functions you can think of that fit these conditions? Thanks. (Note: About a month ago, I asked a somewhat similar question that was sort of, but not exactly — the initial point was f(0) = 0, not f(1) = 1 — the inverse of this question. Thank you for the many responses on that question.)

—SeekingAnswers (reply) 08:09, 23 March 2016 (UTC)


 * Either


 * $$f(x) = c + (1 - c) e^{1 - x}$$


 * which works for all real $$x$$ and $$c$$ &lt; 1 or if you only require the function to be defined for positive $$x$$, then


 * $$f(x) = c + (1 - c) \frac{1 + x_{0}}{x + x_{0}}$$


 * which works for all real $$x$$ &gt; $$- x_{0}$$ and $$c$$ &lt; 1. --catslash (talk) 10:30, 23 March 2016 (UTC)

Any other functions? Do most of the answers from the previous question not translate as well to this one? —SeekingAnswers (reply) 23:00, 24 March 2016 (UTC)


 * Every possible answer to the other question translates to this one if you multiply by -1, and add an appropriate constant and scale as desired. --JBL (talk) 23:06, 24 March 2016 (UTC)