According to the necessary and sufficient conditions of differentiability and the definition of dy,

For differentiable functions, when △x→0

△y=A△x+o(△x)=Adx +o(△x)= dy+o(△x), and o(△x) represents the high-order infinitesimal of △x.

So △y -dy=(o(△x)

(△y -dy)/△x = o(△x) / △x = 0

So it is high-order infinitesimal.

Extended data

A variable in a function can never coincide with a while it is getting bigger (or smaller) and approaching a certain value ("it can never be equal to a, but it is enough to get a high-precision calculation result"). The change of this variable is artificially defined as "always approaching without stopping", and it has a "never equal to a".

The basic method of finding the limit is as follows

1, in the fraction, the numerator and denominator are divided by the highest degree, and infinity is calculated as infinitesimal, and infinitesimal is directly substituted into 0;

2. When the infinite root is subtracted from the infinite root, the molecule is physical and chemical;

3. Apply the Lobida rule, but the application condition of the Lobida rule is that it is transformed from infinity to infinity or infinitesimal to infinitesimal, and the numerator denominator must also be a continuous derivative function.

4. The expansion is based on McLaughlin series, but it is generally misinterpreted as Taylor expansion in China.