  By performing a Taylor expansion along the extra dimension of a metric
describing a black hole on a brane, we explore the influence of the embedding
space on the black hole horizon. In particular, it is shown that the existence
of a Kottler correction of the black hole on the brane, in a viable braneworld
scenario, might represent the radius of the black string collapsing to zero,
for some point(s) on the black string axis of symmetry along the extra
dimension. Further scrutiny on such black hole corrections by braneworld
effects is elicited, the well-known results in the literature are recovered as
limiting cases, and we assert and show that when the radius of the black string
transversal section is zero, as one moves away from the brane into the bulk, is
indeed a singularity.
