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."

"In my view, this definition truly achieves complete 'staticity', without any trace of motion, and leaves nothing unclear."

"Fat Fish, with your intellect, it should not be difficult to see that it doesn't care how you approach L, whether you fly or twist around."

"As long as the final difference is smaller than ε, I acknowledge l as the limit at a."

"For example, consider the simplest f(x)=1/x; as x grows larger, the value of the function becomes smaller: f(1)=1, f(10)=0.1, f(100)=0.01, f(1000)=0.001..."

"...Clearly, as x increases, the value of f(x) approaches 0. Therefore, the limit of f(x) at infinity should be 0."

"Next, take an arbitrarily small ε, say ε=0.1; then find a δ to see if there's a range such that |f(x)-0| is less than 0.1."

"Clearly, x just needs to be greater than 10; for ε=0.01, just make x greater than 100."

"Given any ε, we can obviously find a number that when x is greater than this number, |f(x)-0| is less than ε, and then it's alright."

"What do you think, isn't my idea quite brilliant?"

Several minutes later.

Xu Yun looked up from the letter with admiration.

Although the phrase is cliché,

at this moment, he really wanted to gasp in astonishment at how terrifyingly talented this person is...

As is well known,

the rudiments of calculus can be traced back to a long time ago, with many enlightened minds from different eras proposing related concepts.

For example, Archimedes, Aristotle, Liu Hui, among others.

Building on the work of these pioneers,

in the mid-to-late 17th century, Newton and Leibniz each independently established a systematic calculus.

However, those who truly understand the context know that the calculus created by Newton and Leibniz was not perfect.

Just like Little Niu said, it had a fatal flaw:

The concept of limits was too vague.

Therefore, many tried to patch this flaw, like Maclaurin who tried to explain it with instantaneous velocity, and Taylor with differences.

But from the perspective of later generations, their approaches were clearly not right.

Hence, during this phase,

there were many critiques and doubts about calculus theory.

The most representative was Bishop Berkeley, marking the first mathematical crisis proposed earlier.

To resolve the crisis, what was needed?

The answer is simple: only by making the concept of limits truly rigorous.

Later, through the efforts of D'Alembert, Bolzano, Abel, and Cauchy, they finally defined definite integrals as a sum limit.

Finally, with Weierstrass, a great mathematician, adding the final piece, the now commonly used logically rigorous ε-δ definition of function limits was achieved.

Note that Weierstrass achieved this at the end of the 20th century, two hundred years after Little Niu and his peers created calculus!

Yet in this letter, Little Niu, relying on his own strength, has deduced the concept of limits to its ultimate form!

Indeed,

at that time, Little Niu had Yang Hui's Triangle and Taylor's Formula to assist him, which were entirely different from the historical Little Niu.

But both served only as auxiliary aids, at most helping with the initial steps.

The decisive factor was Little Niu's personal capability.

Looking at the letter in front of him, a thought suddenly surged in Xu Yun's heart:

If Little Niu could come to the modern world like Old Su, how high would his achievements be?

But soon,

Xu Yun shook his head and dismissed the thought.

Old Su's coming to the modern world was very fortuitous and closely related to the historical context.

Achieving the same level in the 1665 Dungeon is extremely challenging.

Although the image of Old Su and Little Niu taking off together... ah no, both excelling in the modern world is beautiful, it's unlikely to happen anytime soon.

Further,

The Halo bringing this letter to me evidently isn't just for Little Niu to report his achievements; there must be other purposes.

Perhaps it's a task, perhaps something else.

With this thought,

Xu Yun lowered his head and read the letter again.

"...The above are my mathematical discoveries; if you are not well-versed in calculus, it might provide you some inspiration."