Page 1: Definition and Basic Concepts of Sequence Limits

This page introduces the fundamental concept of granicą ciągu liczbowego (limit of a numerical sequence). It presents the formal definition of a sequence limit and illustrates it with mathematical notation and examples.

Definition: The limit of a sequence {an} is a value L if for every ε > 0, there exists a positive integer N such that |an - L| < ε for all n ≥ N.

The page showcases various examples of sequence limits, including:

Example: lim(n→∞) 1/n = 0

Example: lim(n→∞) 5 = 5

Example: lim(n→∞) 1/n² = 0

These examples demonstrate how different sequences converge to their respective limits as n approaches infinity. The page also touches on the concept of granicą właściwą funkcji (proper limit of a function) in the context of sequences.

Vocabulary: Przestrzeń metryczna (metric space) - A set where a notion of distance between elements is defined.

The page concludes with a theorem statement, likely related to sequence limits, setting the stage for more advanced concepts in the following pages.

Page 2: Arithmetic Operations with Sequence Limits

This page delves into the rules for performing arithmetic operations with sequence limits, specifically focusing on addition, multiplication, and division. These rules are essential for solving more complex limit problems and understanding the behavior of combined sequences.

Highlight: The reguła dodawania i mnożenia (rule of addition and multiplication) for limits is a fundamental concept in calculus.

The page begins by explaining the addition of limits:

Example: lim(n→∞) (1 + 2) = lim(n→∞) 1 + lim(n→∞) 2 = 0 + 2 = 2

This example demonstrates that the limit of a sum equals the sum of the limits.

Next, the multiplication of limits is discussed:

Example: lim(n→∞) (6/n · 2/n²) = lim(n→∞) 6/n · lim(n→∞) 2/n² = 0 · 0 = 0

This illustrates how the limit of a product is equal to the product of the limits.

The page concludes with an example of limit division:

Example: lim(n→∞) (2n² - 3n) / (n² + 5) = 2

This complex example showcases the application of limit rules to rational functions, demonstrating how to handle indeterminate forms and simplify expressions to find the limit.

Vocabulary: Metryką Manhattan (Manhattan metric) - A way of measuring distance in a grid-like path between two points.

The page provides valuable insights into obliczanie granic ciągów zbieżnych (calculating limits of convergent sequences) and reinforces the importance of these rules in solving advanced limit problems.