L'Hôpital's rule replaces the limit of f/g by the limit of f'/g' - but only for indeterminate forms like 0/0. Why does applying it to an ordinary limit give the wrong answer?

B. Because the rule is not a general identity about quotients.

Test it where the form is not indeterminate: (x+1)/(x+2) at 0 is 1/2 by substitution, and differentiating top and bottom gives 1. The proof runs through the Cauchy Mean Value Theorem and depends on both functions vanishing, so check the form every time, including after each application. It can also fail to help: x/√(x²+1) returns you to a similar expression indefinitely.