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Understanding the limit of a sequence is crucial in mathematics, particularly in analyzing its convergence. A sequence converges to a limit if its terms get arbitrarily close to a certain value as the index increases. This text delves into the formal definition of limits, graphical representations, and the unique properties of sequence limits. It also discusses the application of limit laws, such as the Sum Rule and Quotient Rule, and explores special theorems for divergent sequences.

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## The Concept of a Limit

### Definition of a Limit

A limit is the value that a sequence's terms approach as the index increases indefinitely

### Convergence

Formal Definition of Convergence

A sequence converges to a limit if, given any small positive number, there is a natural number beyond which all terms of the sequence fall within a certain neighborhood of the limit

Divergence

A sequence is considered to diverge if it does not approach a finite limit

### Graphical Representations

Graphs can visually demonstrate the behavior of a sequence as the index increases and confirm convergence to a limit

## Limit Laws

### Sum Rule

The Sum Rule allows for the separation of the limit of a sum into the sum of individual limits

### Product Rule

The Product Rule allows for the combination and manipulation of known convergent sequence limits

### Constant Multiple Rule

The Constant Multiple Rule facilitates the computation of limits for complex sequences

### Quotient Rule

The Quotient Rule allows for the combination and manipulation of known convergent sequence limits, with the caveat that the denominator does not approach zero

## Uniqueness of Sequence Limits

### Proof by Contradiction

The uniqueness of sequence limits can be established through a proof by contradiction, assuming the existence of two distinct limits and demonstrating that this leads to an inconsistency

### Difference Rule

The Difference Rule necessitates that two different limits must be identical, affirming the uniqueness of a sequence's limit

## Divergence of Sequences

### Types of Divergence

Sequences may diverge in various manners, such as increasing without bound

### Special Theorems

Squeeze Theorem

The Squeeze Theorem states that if a sequence is bounded above and below by convergent sequences with the same limit, then it must also converge to that limit

Absolute Value Theorem

The Absolute Value Theorem states that if the limit of the absolute value of a sequence is zero, the sequence itself converges to zero

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