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 3: Comparison Theorem

The final page presents the teorema dei carabinieri or comparison theorem, also known as the squeeze theorem.

Definition: The comparison theorem states that if three functions h$x$, f$x$, and g$x$ satisfy h$x$ ≤ f$x$ ≤ g$x$, and the outer functions have the same limit, then f$x$ must have that same limit.

Vocabulary: "Teorema dei carabinieri" (Two Carabinieri Theorem) is named after the Italian military police, metaphorically "squeezing" the middle function.

Example: The proof demonstrates how the middle function f$x$ is forced to approach the same limit as the bounding functions.

Highlight: This theorem provides a powerful tool for finding limits of functions that are difficult to evaluate directly by comparing them with simpler functions.

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.