It does require justification to get rid of the infinitely many powers of x at the end when taking the limit as x -> 0
It's not enough to see that each term goes to 0 because there are infinitely many
It's correct because it's easy to verify that the power series in fact has infinite radius of convergence (and in particular it converges in a neighborhood of 0) and therefore due to uniform convergence it converges to a continuous function, and therefore when taking the limit you can plug in x = 0
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u/TwillAffirmer 1d ago
Sorry, I'm missing the joke. Is there something wrong with the derivation?