Page 3: Finite and Infinite Limits as x Approaches Infinity

This page covers the behavior of functions as x approaches infinity, discussing both finite and infinite limits in this context.

For finite limits as x approaches infinity, the notation is:

lim f(x) = l x→∞

Definition: A function has a finite limit at infinity if its values approach a specific number l as x grows arbitrarily large.

Highlight: When a function has a finite limit at infinity, the line y = l becomes a horizontal asymptote of the function.

The page then moves on to infinite limits as x approaches infinity, using the notation:

lim f(x) = +∞ or lim f(x) = -∞ x→∞ x→∞

Vocabulary: Limite che tende a infinito esercizi svolti refers to solved exercises involving limits as x approaches infinity, which are crucial for understanding this concept.

The page provides examples of different scenarios, including limits approaching positive infinity, negative infinity, and cases where the limit depends on the direction of approach (x→+∞ or x→-∞).

Example: lim √(x) = +∞ as x→+∞, demonstrating that the square root function grows without bound as x increases. x→+∞

Page 4: One-Sided Limits and Limit Existence

The final page discusses one-sided limits and the conditions for the existence of a limit. It introduces the concepts of right-hand and left-hand limits.

Definition: A right-hand limit is the limit of a function as x approaches a value from the right, while a left-hand limit approaches from the left.

The notation for one-sided limits is:

lim+ f(x) (right-hand limit) x→x₀

lim- f(x) (left-hand limit) x→x₀

Highlight: For a limit to exist, both the left-hand and right-hand limits must exist and be equal.

The page explains that if the one-sided limits are not equal, mathematicians say that the limit does not exist.

Example: An example is provided where lim+ f(x) = 4 as x→1, but lim- f(x) does not exist as x→1, resulting in the non-existence of the overall limit.

Vocabulary: Definizione rigorosa di limite encompasses the formal definition of limits, including the concepts of one-sided limits and limit existence.

This page emphasizes the importance of considering both sides when evaluating limits, especially for functions with discontinuities or different behaviors on either side of a point.