In a book by Wise and Manohar, *Heavy Quark Physics* (pg 80), they discuss the limit

\begin{equation}\lim _{\lambda\rightarrow \infty} \lambda^{\,z\,(\epsilon)}\end{equation}

where $z$ is some function of an infinitesimal parameter, $\epsilon$. Then they say "as long as $z$ depends on $\epsilon$ in a way that allows one to analytically continue $z$ to negative values" then this limit is zero. I'm not very familiar with analytic continuation (other then the qualitative idea of what it means) but this seems very strange to me.

I understand why the paths of contour integrals can be morphed between one another (due to Residue theorem), but why should such arguments hold for limits as well?