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Infinitely small is not zero
There is a clear difference between infinitely small and zero. It has no particular value and you must take care of this during calculations.
Explanation
We call this infinitely small value Δx, and keep in mind that Δx→0. Thus applies
as Δx is neglectable. However, Δx has a value, so you may use it as a divisor. In the further investigation we use the formula
and calculate, with a blind eye, in a first step
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That is however wrong. You cannot do that. We try again, in the proper way, and start with n = 0. For every number a ≠ 0 applies a0 = 1. This time the formula gives
It looks quite clear now. The result is the same in both calculations. We continue with n = 1 and find
That is strange, because that is another result from our calculation with the blind eye. Let's try with n = 2 and we see
Once more, now with n = 3 and get
Apparently is (1 + Δx) smaller than (1 + Δx)2 and that is again smaller than (1 + Δx)3. That could be an explanation. But that cannot be true. We have no doubt that
Moreover we just saw that
and thus every exponent, let us just call it n, must of course give 1 as result, because generally applies
What has gone wrong? Alas, here you will notice that infinitely small is different from zero. The big mistake was made from the very beginning, as it should read
because it is really a limit that we handle here. Special rules apply when processing them. You should never apply a limit to a part of a calculation. So, an error was made, because
and that had fatal consequences, as we have seen.