Fix $q>1$. Define the function 


$$
f\_q(c):=\int\_e^\infty \frac{e^{-c r^2}r}{\log(r)^q}d r.
$$


The problem is whether the following is true, 


$$
\lim\_{c\rightarrow 0} c \log(1/c)^q f\_q(c) = C \in (0,\infty)?
$$


If not, what is the right rate that $f\_q(c)$ blows up at $c=0$?


Note that if $q=0$, then $f\_q(c)=\frac{e^{-c e^2}}{2c}$ and the above limit is true with $C=1/2$.


It is hard to choose right tags for this question, I am sorry for anything improper. 



A calculus question
It converges for all $q$. You can separate the interval into the ranges $[e,1/\sqrt c]$ and $[1/\sqrt c,\infty)$. On the first range, the exponential term is essentially 1. On the second range, the logarithm is essentially constant. 


