In the last years, many results were presented regarding the calculus of limiting subdifferentials. Most of these results were formulated for finite-dimensional spaces. To extend these results to infinite-dimensional spaces, some additional assumptions (Lipschitz conditions, compactness conditions) are necessary. So the question arises, if these assertions remain true also without such conditions. In the present paper, the authors present some counterexamples which demonstrate that the answer is negative. In detail, the following assertions are discussed: \hskip 17mm \(*\) sum rule and chain rule for limiting subdifferentials, \hskip 17mm \(*\) Lagrange multiplier rule for constraint optimization problems, \hskip 17mm \(*\) sum rule and chain rule for coderivatives, \hskip 17mm \(*\) extremal principle of Mordukhovich, \hskip 17mm \(*\) open mapping theorem and metric regularity. All the examples are constructed for separable Banach spaces with convex functions and convex sets, respectively. So these examples can be used also for the discussion of more special types of subdifferentials. At the end of the paper, the authors give some remarks to create more general (nonconvex) examples for more general spaces.