application of L'Hopital's rule? I am trying to evaluate the following  limit: 
$$
\lim_{x \to 0} \frac{e^x}{\sum_{n = 1}^\infty n^k e^{-nx}},
$$
where $k$ is a large (but fixed) positive integer. I am unsure how to proceed. Can this be done using L'Hopital's rule? Just started learning calculus, thanks guys!!
For each $x\lt\frac12$, there is an $n\in\mathbb{N}$ so that $1\lt\frac1x-1\le n\lt\frac1x$. Picking this one term out gives
$$
\sum_{n=0}^\infty n^ke^{-nx}\ge \left(\frac1x-1\right)^ke^{-1}
$$
As $x\to0^+$, $\left(\frac1x-1\right)^ke^{-1}\to\infty$
You don't need L'Hopital's rule; try substituting $0$ for $x$ and see what you get.