Since we're talking about circularity of arguments, here's something that I see that almost every school textbook gets wrong. More specifically, it's the corresponding angle theorem, which states that the corresponding angles in two parallel lines cut by a straight line.
Pretty much every textbook I've seen either just states it or uses completely circular arguments to prove it. In fact, the only rigorous treatment I've seen of it is in Euclid's Elements!
Of course, proofs in geometry can be quite different depending on the postulates you use, but it's still a "hard" proof that I think textbooks should not omit, because it's a key point in Euclidean geometry.
In any case, I found Euclid's proof ingenious. First, he
proves that an exterior angle of a triangle is always greater than the two other internal ones. Then, he uses that to
show that the sum of two angles of a triangle is always less than 180 degrees.
Then, he uses this fact to
demonstrate that when corresponding angles are equal, the lines are parallel, it's remarkable that this is true even without the parallel postulate!
The corresponding angle theorem is the converse of this statement, and Euclid's parallel postulate is written in a form very convenient to prove this theorem, which Euclid then uses to derive a proof by contradiction.
I think this is actually another case of textbooks handwaiving some things and leaving interesting mathematics behind, in this case the necessity of the parallel postulate to prove fundamental statements in geometry, something that was known very well by the Greeks!
which simplifies to
(*)
Setting equal to C (the goal) and solving for n, we get:
(**)
For the example that you gave, plugging in C=1000000, A=2000, x=0.01 gives approximately 180.07 months, so it takes 181 months (~15.08 years) to reach the goal.
But that's not your actual question. The actual question is to prove that the formula tends to C/A when x->0. We could use L'Hopital's rule on (**).
Or instead we could just see that (*) approaches An as x->0. This is because ((1+x)^n - 1) / (1+x - 1) is precisely the value of the derivative of z^n at z=1, by definition. This gives An=C, so it follows that n=C/A in the case x->0.
