Page 2: Sign Permanence Theorem

This page covers the teorema della permanenza del segno and its inverse, explaining how the sign of a function relates to its limit.

Definition: The sign permanence theorem states that if the limit of a function is non-zero, the function maintains the same sign as its limit in some neighborhood of the limit point.

Vocabulary: "Permanenza del segno" translates to "permanence of sign," referring to the consistent behavior of a function's sign near its limit point.

Example: For a positive limit l, there exists a neighborhood where f(x) remains positive; similarly for negative limits.

Highlight: The theorem specifically excludes the case where the limit equals zero, as sign behavior cannot be guaranteed in this case.

Page 1: Uniqueness of Limit Theorem

The first page introduces the teorema di unicità del limite, establishing that if a function has a finite limit at a point, that limit must be unique. The proof employs contradiction methodology.

Definition: The uniqueness theorem states that if a function f(x) has a limit as x approaches x₀, then this limit must be unique.

Example: The proof assumes two different limits l and l' exist, leading to a contradiction through epsilon-based inequalities.

Highlight: The proof cleverly uses the epsilon-delta definition of limits to show that assuming two different limits leads to a mathematical contradiction.