Consider the following erroneous usage of L'hopital's rule:
$$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{D_h(f(x+h) - f(x))}{D_h(h)} = \lim_{h \to 0} \frac{f'(x+h)}{1} = f'(x) \tag1 $$
Note that if $\lim_{h \to 0} f'(x+h)$ exists, it is always equal to $f'(x)$.
There are two remarks one can make about $(1)$. The usage of L'hopital's rule is entirely fallacious, since we're essentially using a rule which uses the concept of differentiation on the definition of the derivative itself, and that, notwithstanding this, its usage yields the correct answer. Note also that $(1)$ is essentially a generalization of the common (explicit) example of this fallacy where the limit is $\frac{\sin x}{x}$.
My question is twofold:
Is the fact that the circular reasoning above gave us the correct answer entirely a mathematical coincidence, or is there some "deeper" reason behind it? In either case, is there an example in mathematics, analagous to $(1)$, where circular reasoning gives an incorrect answer?