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We study higher order tangents and higher order Laplacians on fully symmetric p.c.f. self-similar sets with three boundary points. Firstly, we prove that for any function f defined near a vertex x, the higher order weak tangent of f at x, if exists, is the uniform limit of local multiharmonic functions that agree with f near x in some sense. Secondly, we prove that the higher order Laplacian on a fractal can be expressible as a renormalized uniform limit of higher order graph Laplacians. Somedoi:10.4064/sm191017-24-5 fatcat:jgkpbjk2qnhorm4kort56rj2lq