The first proof in your diagram can be correct, because L'H?pital is not needed in the last step, only E X- 1 and X are equivalent infinitesimal. Looking at the advanced mathematics textbooks carefully, the logic should be: take the limit of (1+x) (1/x) as e (the derivative is not involved in the proof process), prove that ln(x+ 1) and X are infinitely equivalent, and then prove that E X- 1 and X are infinitely equivalent.

Prove the basic limit of infinite power of 1

Solve the limit by using the infinite power basic limit of 1