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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 Δ*x*, and keep in mind that Δ*x*→0. Thus applies

as Δ*x* is neglectable. However, Δ*x* 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

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 *a*^{0} = 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* +* Δ*x*) smaller than (1* +* Δ*x*)^{2} and that is again smaller than (1* +* Δ*x*)^{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.