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As Rbega says in the comments, if you are really keen to see this rescaling idea put to use in a more rigorous or advanced way, then you can look at some Geometric Measure Theory. While it will look very technical compared to this (because it is designed for potentially badly-behaved or very weakly-defined geometric objects), this sort of homothetic blowing up is standard for defining tangent objects to things. You get a weak kind of convergence of the rescalings of your original object to the tangent object, which, depending on the circumstances, may well (or perhaps will hopefully) display some sort of rigidity, e.g. it may be have to be a cone. It is rigorous and yes you can indeed end up with things like the union of two lines as your tangent object.

In the special case of the graph of a differentiable function, the tangent object at a point will indeed be the graph of the affine function associated with the derivative at the point.

I don't know of any books which take this approach pedagogically, in the development of calculus though.

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As Rbega says in the comments, if you are really keen to see this idea put to use, then you can look at some Geometric Measure Theory. While it will look very technical compared to this (because it is designed for potentially badly-behaved or very weakly-defined geometric objects), this sort of homothetic blowing up is standard for defining tangent objects to things. You get a weak kind of convergence of the rescalings of your original object to the tangent object, which, depending on the circumstances, may well (or perhaps will hopefully) display some sort of rigidity, e.g. it may be have to be a cone. It is rigorous and yes you can indeed end up with things like the union of two lines as your tangent object.

In the special case of the graph of a differentiable function, the tangent object at a point will indeed be the graph of the affine function associated with the derivative at the point.

I don't know of any books which take this approach pedagogically, in the development of calculus though.