  We obtain sufficient conditions under which the limit of a sequence of
functions exhibits a particular dynamical behaviour at a point like
expansivity, shadowing, mixing, sensitivity and transitivity. We provide
examples to show that the set of all expansive, positively expansive and
sensitive points are neither open nor closed in general. We also observe that
the set of all transitive and mixing points are closed but not open in general.
We give examples to show that properties like expansivity, sensitivity,
shadowing, transitivity and mixing at a point need not be preserved under
uniform convergence and properties like topological stability and
$\alpha$-persistence at a point need not be preserved under pointwise
convergence.
