I have a function with the following behavior: $$ f(0) = 1\\ \lim_{x\to\infty}f(x) = \alpha $$ Here, $\alpha$ is a positive constant (say, 2). The function $f$ is monotonically increasing and asymptotically approaching $\alpha$. The plot between $l := \log(x)$ and $f$ is like a sigmoid function. I tried fitting $f$ with the standard sigmoid functions of $l$, like logisitic, error, tan hyperbolic, arctan functions but nothing works good [The RMS error is significant with respect to variation in $f$]
See the graphs for reference.
f as a function of log x, f as function of x
Is there anything else I can do?
See the fitting below.
The blue curve is the curve fitted to a set of points (red) taken by scanning your figure entitled " f as a function of log x ". Thus $X=$log x .
You wrote log x . If it is the Natural logarithm (base e) then the above formula is equivalent to $$Y=a+\frac{b}{1+\gamma\:x^{-\delta}}\quad\text{with}\quad X=\ln(x)\quad;\quad\gamma=e^{c/d} \quad ;\quad \delta=1/d$$ If your log is another logaritm (different base than e) convert it first to put it into the formula.
NOTE : The data used above is :
X = -13.41 , -11.36 , -10.95 , -9.69 , -8.99 , -8.01 , -7.07 , -6.05 , -5.35 , -4.46 , -3.8 , -3.02 , -2.33 , -1.8 , -1.02 , -0.37 , 0.08 , 0.53 , 1.06 , 1.63 , 2.21 , 2.86 , 3.39 , 3.84 , 4.62 , 5.19 , 5.97 , 6.54 , 7.64 , 8.54 , 9.36 , 10.18 , 10.91 , 11.69 , 12.75 , 13.16
Y = 1.002 , 1.005 , 1.005 , 1.007 , 1.009 , 1.014 , 1.021 , 1.032 , 1.041 , 1.062 , 1.08 , 1.107 , 1.139 , 1.167 , 1.217 , 1.265 , 1.299 , 1.34 , 1.386 , 1.438 , 1.486 , 1.543 , 1.584 , 1.614 , 1.659 , 1.687 , 1.716 , 1.732 , 1.755 , 1.767 , 1.773 , 1.778 , 1.78 , 1.783 , 1.785 , 1.785