  While finite graphs have tree-decompositions that efficiently distinguish all
their tangles, locally finite graphs with thick ends need not have such
tree-decompositions. We show that every locally finite graph without thick ends
admits such a tree-decomposition, in fact a canonical one. Our proof exhibits a
thick end at any obstruction to the existence of such tree-decompositions and
builds on new methods for the analysis of the limit behaviour of strictly
increasing sequences of separations.
