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. Some

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... sults can be extended to general p.c.f. self-similar sets. In the Appendix, we provide a recursive algorithm for the exact calculations of the boundary values of the monomials on D3 symmetric fractals, which is shorter and more direct than in the previous work on the Sierpiński gasket.