Definition

Noun: 1. A mathematical value approached by a function: The value that a function f(x) gets arbitrarily close to as the independent variable x approaches a specific value (often infinity or a particular point c). It is the value that the sequence or function "tends to" or "converges to."

Usage

The term "limit point" is used in mathematical analysis, particularly in calculus and real analysis, to describe the behavior of sequences and functions. * It is the central concept in defining limits, continuity, derivatives, and integrals. * It describes the idea of approximation and convergence.

Examples

• For a sequence: The sequence 1, 1/2, 1/3, 1/4, ... has a limit point of 0.
• For a function: As approaches 0, the function has a limit point of 1.
• In a proof: "To prove the function is continuous, we must show that for every sequence converging to , the corresponding sequence of function values converges to , which is the limit point."

Advanced Usage

• In topology: The term has a more general definition. A point is a limit point (or accumulation point) of a set if every neighborhood of contains at least one point of different from itself. This means points of get arbitrarily close to .
  • Example: For the set S = {1/n : n is a natural number}, the point 0 is a limit point of S, even though 0 is not itself in S.

Variants and Related Words

• Limit (n): Often used synonymously with "limit point" in the context of functions and sequences. ("The limit of the function is 5.")
• Limiting value (n): A synonym for limit point.
• Point of convergence (n): The value where a sequence or series converges.
• Accumulation point (n): A synonym used primarily in topology and real analysis.

Synonyms

• Limit
• Limiting value
• Accumulation point (in topological contexts)

Related Concepts (Not Phrasal Verbs)

• To approach a limit: The process of a variable getting closer to a limit point.
  • The terms of the sequence approach the limit of zero.
• To converge to: To have a specific limit point.
  • The infinite series converges to the value of π/4.
• To tend to: Another way to say "approach."
  • As x tends to infinity, f(x) tends to a constant.

Noun

1. the mathematical value toward which a function goes as the independent variable approaches infinity