But for an infinite hotel, it is possible for both to be true at the same time (indeed the second statement is always true for an infinite hotel). Put another way: for finite hotels, we use "full" to mean "there's a guest in every room", and we also use "full" to mean "they can't fit in another guest".
set theory - Hilbert's Grand Hotel is always hosting the same infinite ...
I am a little confused about how a cyclic group can be infinite. To provide an example, look at $\langle 1\rangle$ under the binary operation of addition. You can never make any negative numbers with
sequences and series - What is the sum of an infinite resistor ladder ...
This resolves your problem because it shows that $\frac {1} {\epsilon}$ will be positive infinity or infinite infinity depending on the sign of the original infinitesimal, while division by zero is still undefined. This viewpoint helps account for all indeterminate forms as well, such as $\frac {0} {0}$.
There are the following textbooks to learn about infinite-dimensional manifolds: "The Convenient Setting of Global Analysis" by Andreas Kriegl and Peter W. Michor
functional analysis - What is a good textbook to learn about infinite ...
The dual space of an infinite-dimensional vector space is always strictly larger than the original space, so no to both questions. This was discussed on MO but I can't find the thread.
linear algebra - What can be said about the dual space of an infinite ...