The definition of function limit in advanced mathematics is described by "ε-δ" language, for example, the definition of limit of function f(x) at x0: ε >; 0, with δ > 0, so when 0

This definition is simple: if it conforms to the "ε-δ" language, the limit of the function is a.

Note: this definition is also correct in reverse: if "the limit of f(x) in x0 is a", then "take ε >; 0, with δ > 0, so when 0

In the process of proving the "limit algorithm of composite function", this definition is actually used repeatedly, with positive and negative.

Proving the limit of composite function is equivalent to proving this proposition: take ε >; 0, with δ > 0, so when 0

Start to prove:

Because lim(u→u0)f(u)=A, ε >; 0 with η >; 0 when 0 < | u-u0 | < When η, | f (u)-a | < ε holds-①.

And because lim(x→x0)g(x)=u0, for η>:0, there is δ1>; 0, so when 0

These two sentences use the definition of function limit in turn: if the function limit is a, it conforms to the "ε-δ" language.

η appearing in ② has the same meaning as ε in "ε-δ" language, and both represent infinitesimal numbers, which must be greater than 0 in the definition of function limit. Why write different letters when it also means infinity?

The key reason is that different functions will use different ε (infinitesimal) when proving the limit of a function with "ε-δ" language. ① ② It is aimed at different functions, so infinitesimal needs to be represented by different letters.

For question (2), "Suppose ...", how does the following "| f [g (x)]-a | = | f (u)-a | < ε hold"?

"Assumption ……" The assumption here is: the precondition of the limit algorithm of composite function.

I can't write correctly. I read it in a book.