  The notion of topological entropy can be conceptualized in terms of the
number of forward trajectories that are distinguishable at resolution
$\varepsilon$ within $T$ time units. It can then be formally defined as a limit
of a limit superior that involves either covering numbers, or separation
numbers, or spanning numbers. If covering numbers are used, the limit superior
reduces to a limit. While it has been generally believed that the latter may
not necessarily be the case when the definition is based on separation or
spanning numbers, no actual counterexamples appear to have been previously
known. Here we fill this gap in the literature by constructing such
counterexamples.
