limit point: meaning, definition, pronunciation and examples
C2+ (Very High Proficiency, Specialized)Highly technical, academic
Quick answer
What does “limit point” mean?
A point that can be approximated arbitrarily closely by other points in a given set, excluding itself.
Audio
Pronunciation
Definition
Meaning and Definition
A point that can be approximated arbitrarily closely by other points in a given set, excluding itself.
In mathematics, specifically in topology and real analysis, a point where every neighbourhood contains at least one point from a given set distinct from the point itself. Also called an 'accumulation point' or 'cluster point'.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or usage differences; the term is identical in both mathematical traditions.
Connotations
Purely mathematical, with no additional cultural connotations in either variety.
Frequency
Used with identical frequency in advanced mathematics textbooks and lectures in both regions.
Grammar
How to Use “limit point” in a Sentence
The limit point of [SET][SET] has [NUMBER] limit points[POINT] is a limit point of [SET]to find/identify/calculate the limit pointVocabulary
Collocations
Examples
Examples of “limit point” in a Sentence
verb
British English
- [No standard verb form]
American English
- [No standard verb form]
adverb
British English
- [No standard adverb form]
American English
- [No standard adverb form]
adjective
British English
- The limit-point concept is fundamental.
- We studied limit-point criteria.
American English
- The limit-point concept is fundamental.
- We studied limit-point criteria.
Usage
Meaning in Context
Business
Not used.
Academic
Central term in undergraduate and graduate-level real analysis, topology, and complex analysis courses.
Everyday
Virtually never used in everyday conversation.
Technical
The primary context. Precision is critical; used in proofs, definitions, and problem statements.
Vocabulary
Synonyms of “limit point”
Strong
Weak
Vocabulary
Antonyms of “limit point”
Watch out
Common Mistakes When Using “limit point”
- Confusing 'limit of a sequence' with 'limit point of a set'.
- Assuming every point in a set is its own limit point (false; isolated points are not limit points).
- Using 'limit point' in non-mathematical contexts where 'limit', 'boundary', or 'cap' would be appropriate.
FAQ
Frequently Asked Questions
No. A limit point of a set can be inside the set or on its boundary. The key is that it can be approximated by other points from the set.
The limit of a sequence is a specific value the sequence approaches. A limit point of a set is a point that can be approached by infinitely many distinct points from the set, but not necessarily in a specific order.
No. In a metric space (like the real numbers), a finite set has no limit points because you can put a small enough neighbourhood around any point that excludes all other points of the finite set.
In standard real analysis and topology, yes, these terms are used interchangeably to mean the same precise concept.
A point that can be approximated arbitrarily closely by other points in a given set, excluding itself.
Limit point is usually highly technical, academic in register.
Limit point: in British English it is pronounced /ˈlɪmɪt ˌpɔɪnt/, and in American English it is pronounced /ˈlɪmɪt ˌpɔɪnt/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “[No common idioms for this technical term]”
Learning
Memory Aids
Mnemonic
Think of a limit point as a 'magnet point' – other points in the set are irresistibly drawn close to it, infinitely often.
Conceptual Metaphor
A SPATIAL ATTRACTION metaphor: a point that 'attracts' or 'collects' other points infinitely close to it.
Practice
Quiz
Which of the following is NOT necessarily true about a limit point p of a set S?