  In this work we present a model-independent search strategy at the LHC for
heavy Higgs bosons decaying into a tau and a muon, $H/A \rightarrow \tau\mu$,
showing a plausible tendency to improve the sensitivity obtained by the present
experimental limits. This search strategy is performed for the Higgs boson mass
range 1-5 TeV and uses as the most relevant kinematical variables, in order to
discriminate signal against background, the transverse momenta of the muon and
the tau together with the missing transverse energy. We estimate the exclusion
limits at 95% C.L. and the significances for evidence and discovery at
$\sqrt{s}$ = 14 TeV with $\cal{L}$ = 300 fb$^{-1}$, observing a growth in the
sensitivities for high Higgs boson masses. Moreover, since the Higgs boson
decay into a $\tau$-lepton pair may mimic our LFV signal, we also study the
impact of the ditau channel on the exclusion limits and the significances for
evidence and discovery. In particular, the impact on the exclusion limits of
LFV heavy Higgs boson decays is significant when the ditau rate begins to
compete with the corresponding to the $H/A \rightarrow \tau\mu$ decay.
