Physical examples where changing the order of limits yields wrong result In mathematics it is generally not allowed to change order of limits. For example it is not always true for a sequence of functions $f_n$, that $\int_a^b \left(\sum_{n=0}^\infty f_n(x) \right) dx = \sum_{n=0}^\infty \left(\int_a^b f_n(x) dx\right)$. (Note that series $\sum_{n=0}^\infty\ldots$ and the integral $\int_a^b \ldots dx$ are mathematically defined via limits of sequences).
In my experience it happens a lot in physics lectures, that limits are changed in their order without any additional comment (such as mentioning Fubini's theorem or uniform convergence). It also seems to me that there are not many examples relevant for physics where changing the order of limits yield wrong results.
I'm looking for good physical examples showing to students that one has to be careful when he changes the order of limits. So for which physical example the order of the limits is important and you get a wrong result, when you change it? 
 A: The low-frequency ($\omega\rightarrow 0$), long-wavelength ($q\rightarrow 0$) conductivity of an electron gas in the random phase approximation depends on the order in which those two limits are taken. 
Intuitively, if you take the $\omega\rightarrow 0$ limit first, you're talking about a static, long-wavelength potential to which the electrons adjust, so the conductivity is imaginary (non-dissapative).  If you take the $q\rightarrow 0$ limit first, you're talking about a uniform, slowly varying field applied to the gas, so you get a current (real conductivity).  See page 27 of these notes for a discussion.
Basically, you order the limits one way to learn about Thomas-Fermi screening, and the other way to learn about DC currents.  Changing the limits here is not "wrong" per se, but it may get you an answer to a question you didn't ask!
A: For example, in statistical mechanics, you get different results for systems with spontaneous symmetry breaking, say, for a ferromagnetic, depending  on whether you first take the limit $N\rightarrow\infty$ or $H\rightarrow 0$ when calculating average magnetization (http://www.encyclopediaofmath.org/index.php/Quasi-averages,_method_of ).
