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

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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 $c$ < 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 $x$ > $-x_0$ and $c$ < 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)