>>14015642i once discovered a new (to me) proof for eulers identity while thinking about roots of unity. the proof involved substituting small angle approximations (sin x = x, cos x = 1). i wondered how i could justify the rigor of this step, and i came to the conclusion that as x approached zero i could replace the functions with any other function that intersected it at zero and shared its first derivative at zero. this is how i discovered taylor series'.
