Help with the following limit

Question

I have the following limit

$$ \lim_{u\to\infty} \frac{-bu^{a-1}e^u}{e^{be^u}} $$

where a and b are constants

I have tried L'Hopital and i keep getting undefined results. I have also tried Series expansions but nothing.

Could anyone take me through this , and how i am supposed to think when tackling a problem such as this?

Thank you very much for your help and time.

Answer 1

First suppose $b>0$. For large $u$ we'll have $be^u>2u$, so $$\frac{u^{a-1}e^u}{e^{be^u}}<\frac{u^{a-1}e^u}{e^{2u}}=\frac{u^{a-1}}{e^u}\to0\ .$$ If $b\le0$ then $e^{be^u}\le1$ so $$\frac{u^{a-1}e^u}{e^{be^u}}\ge u^{a-1}e^u\to\infty\ .$$