I'm reading Henle/Kleinenberg's Infinitesimal Calculus. They say on page $8$:
Can you provide examples in which it is very difficult to find a $\delta$? The example they provided in the previous page is very elementar, I'd like to see how hard it could be.
The first thing that comes to my mind is the Riemann-Zeta function $\zeta(s)$:
$$ \zeta(s) = \sum_{n=1}^{\infty} n^{-s} = 1 + \frac{1}{2^s} + \frac{1}{3^s}+\dots, \quad s > 1$$ without using any identitites, this sure looks complicated to me. I guess you could easily come up with other examples of function series that are difficult to handle.