Limit Definition and 999 Threads

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.

The limit inferior of a sequence




x

n




{\displaystyle x_{n}}
is denoted by





lim inf

n





x

n




or





lim
_



n






x

n


.


{\displaystyle \liminf _{n\to \infty }x_{n}\quad {\text{or}}\quad \varliminf _{n\to \infty }x_{n}.}
The limit superior of a sequence




x

n




{\displaystyle x_{n}}
is denoted by





lim sup

n





x

n




or





lim
¯



n






x

n


.


{\displaystyle \limsup _{n\to \infty }x_{n}\quad {\text{or}}\quad \varlimsup _{n\to \infty }x_{n}.}

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  1. Y

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  3. D

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  4. Mr Davis 97

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  5. J

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  6. M

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  7. Mr Davis 97

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  8. K

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  9. A

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  10. Y

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  11. M

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  12. mbrmbrg

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  13. karush

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  14. F

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  15. T

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  16. evinda

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  17. evinda

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  18. K

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  19. Mr Davis 97

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  20. karush

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  22. opus

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  26. karush

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  30. V

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  31. V

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  34. C

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  39. A

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  40. V

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  41. M

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  42. gibberingmouther

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  43. MountEvariste

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  44. E

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  47. D

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  48. EEristavi

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