I am trying to conceptualize a curve resulting from feeding a ratio between two numbers into a sigmoidal function, to constrain the resulting outputs between 0 and 1.
$$\frac{1}{1 + e^{-x}}$$
I know that when the two numbers in the denominator and numerator are the same the ratio will be 1 and feeding 1 through the sigmoidal function will give me 0.73. How does the function behave as the two numbers in the ratio diverge in the two respective directions (i.e. higher numerators than denominators and vice versa)?
What I am asking for here is an intuitive explanation. A simulation is not required, but would greatly appreciated, as I lack the technical skills to create one myself.
Let $x$ be the ratio of two positive numbers, then $$ \lim_{x\to\infty}\frac{1}{1+e^{-x}}=1, $$
and
$$ \lim_{x\to 0^+}\frac{1}{1+e^{-x}}=1/2. $$ You can type in google search engine "plot y=1/(1+exp(-x))" and you will see the plot.