Please forgive my vocabulary & usage because I'm only a math amateur, so I'll try to describe this the best I can.
Does such a function exist that has an infinite order derivative with a constant value, an initial value of 0, and a final value of infinity.
If so, what is it?
I know this is probably an incorrect way to describe a function, but I am unsure how to properly describe this function.
It's graph resembles geometric growth.
Could it look something an infinite number of integrals of $(1+x)^n$?
The classic function with all derivatives zero that doesn't stay stuck at zero is $e^{\frac {-1}x}$ near zero but positive, $0$ for $x$ negative. That doesn't go to infinity for large $x$, but that is easily fixed. How about $f(x)=xe^{{-1}/x}$ for positive $x$, $0$ for $x \le 0$