  Noncommutative geometry provides a framework, via the construction of
spectral triples, for the study of the geometry of certain classes of fractals.
Many fractals are constructed as natural limits of certain sets with a simpler
structure: for instance, the Sierpi\'nski is the limit of finite graphs
consisting of various affine images of an equilateral triangle. It is thus
natural to ask whether the spectral triples, constructed on a class of fractals
called piecewise $C^1$-fractal curves, are indeed limits, in an appropriate
sense, of spectral triples on the approximating sets. We answer this question
affirmatively in this paper, where we use the spectral propinquity on the class
of metric spectral triples, in order to formalize the sought-after convergence
of spectral triples. Our results and methods are relevant to the study of
analysis on fractals and have potential physical applications.
