Page 2: Infinite Limits for x Approaching a Finite Value

This page discusses the concept of infinite limits as x approaches a finite value. It introduces the notation and definition for both positive and negative infinite limits.

Definition: An infinite limit occurs when the function values grow without bound as x approaches a specific value.

The page presents the notation for infinite limits:

lim f(x) = +∞ or lim f(x) = -∞ x→x₀ x→x₀

Highlight: For an infinite limit, we show that for any large positive number M, there exists a δ > 0 such that f(x) > M $or f(x) < -M for negative infinity$ whenever |x - x₀| < δ.

An example is provided to illustrate the verification of an infinite limit:

lim $x-1$² = +∞ x→1

Example: The verification process involves finding a suitable δ for any given M > 0, such that $x-1$² > M when |x-1| < δ.

The page also introduces the concept of vertical asymptotes, which are related to infinite limits.

Vocabulary: Limite infinito per x che tende a un valore finito describes the situation where a function approaches infinity as x approaches a finite value, often resulting in a vertical asymptote.

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.

Page 1: Introduction to Limits and Finite Limits

This page introduces the concept of limits and focuses on finite limits as x approaches a finite value. It provides a formal definition and examples of limit verification.

Definition: A limit is the value that a function approaches as the input (usually x) gets closer to a specific value.

The page explains the notation for limits:

lim f(x) = l x→x₀

This means that as x approaches x₀, f(x) approaches l.

Highlight: The formal definition of a limit involves the use of epsilon (ε) and delta (δ) to describe the behavior of the function near the limit point.

An example is provided to verify the limit:

lim $2x+1$ = 7 x→3

Example: The verification process involves showing that for any ε > 0, there exists a δ > 0 such that |f(x) - l| < ε whenever |x - x₀| < δ.

The page concludes with another example of limit verification, demonstrating the step-by-step process for a more complex function.

Vocabulary: Limite di una funzione definizione refers to the formal definition of a limit of a function, which is crucial for understanding the concept rigorously.