We have the next function: $f=\frac{q}{\sqrt{1-qt}}$, where $q$ is constant
We want to know that is there some $a\in\mathbb{N}$ which satisfies the equation: $\lim _{n\rightarrow a}\frac{d^nf}{dt^n}=c{,}\ $ where $c$ is constant function and $a\in\mathbb{N}\ or\ a=\infty$?
So we want to find that what is the $c$ or doesn't the given limit approach any constant function?
I tried the next way: we assume that there are some $c$ which depends only on q. We assume that $a\in\mathbb{N}$. Next step is to say that if this is true we get that $f^{\left(a\right)}=c$ and therefore we could find the function $f$ by integrating with $t$. Because we have in the rigth-handside only some constant $c(q)$ we would get next after integration:
$f=k+pt+ot^2+it^3+...+wt^a$
and we know that, $f=\frac{q}{\sqrt{1-qt}}$ therefore we'd get $q^2=\left(k+pt+ot^2+it^3+...+wt^a\right)^{^2}+qt\left(\left(k+pt+ot^2+it^3+...+wt^a\right)^{^2}\right)$
I'm interested in more general case that are there some general way to solve this kind of problems?
The given problem came up when I researched the some kinematic problem. I found that it is hard to accurately say that how much I should take advance with my shotgun when I'm firing ducks which go straight line above me.