Then he glanced at the additional formulas provided by Little Niu:
[If f′(x0) exists, around x0 there is f(x0+Δx)−f(x0) ≈ f′(x0)Δx.]
[Since Δx=x−x0, we can derive f(x)=f(x0)+f′(x0)(x−x0)+o(x−x0).]
[Approximately, we have f(x) ≈ f(x0)+f′(x0)(x−x0).]
This is a very basic differentiation formula, not much different from what Little Niu established historically.
However, while pondering,
Xu Yun suddenly paused, with his expression gradually becoming serious:
"But during the derivation process, I suddenly discovered a problem."
"That is, concepts like 'infinitesimals', 'infinitely close to', dx, seem very vague, sometimes being zero and sometimes not, inevitably causing confusion."
"So, I spent another two and a half years and finally derived a more rigorous mathematical concept."
"If and only if for any ε, there exists a δ so that as |x-a| approaches δ, |f(x)-L| is less than ε."
"Then we say the limit of f(x) at point a is L, denoted as: Limx→af(x)=L."
