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. Math Amateur

    I Limit Points & Closure in a Topological Space .... Singh, Theorem 1.3.7

    I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.2: Topological Spaces ... I need help in order to fully understand Singh's proof of Theorem 1.3.7 ... (using only the definitions and results Singh has established to...
  2. S

    I Proof that limit of difference of infinite limits is indeterminate

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  3. MathematicalPhysicist

    A Symmetric limit in Peskin's and Schroeder's (page 655)

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  4. M

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  5. G

    I Finding an infinitesimal limit function

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  6. PeterDonis

    A What is the Corrected Heisenberg Limit for Phase Estimation Measurements?

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  7. bagasme

    B Rationale Behind t-Substitution for Evaluating Limits?

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  8. Witcher

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  9. Merlin3189

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  10. M

    MHB Check existence of limit with definition

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  11. Z

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  12. T

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  13. Wi_N

    This Limit problem seems too simple....

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  14. archaic

    B Does the Limit Hold for f'(x)>g'(x) Even if f(x)/g(x) is not a Constant?

    Would this ##f'(x)>g'(x)\,\forall x\in [a,\infty)\text{ and }f,\,g\underset{\infty}{\to}0\Rightarrow \lim_{x\to\infty}f(x)/g(x)=0## hold if ##\frac{f(x)}{g(x)}\neq c##?
  15. Y

    MHB Limit involving a hyperbolic function

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  16. Y

    MHB Limit of integer part function using Sandwich rule

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  17. Addez123

    Limit when x^2 + y^2 -> inf, am I solving it correctly?

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  18. J

    MHB What is the solution to the exponential series limit problem?

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  19. crises

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  20. J

    MHB How to fully solve this limit evaluation using integration?

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  21. JTorn

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  22. F

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  23. Amartansh

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  24. Wi_N

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  25. N

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  26. Adesh

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  27. T

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  28. snatchingthepi

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  29. H

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  30. V

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  31. S

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  32. L

    MHB Converging to Delta(y-b): Solving the Limit of f_x((y-b)/a) as a Approaches 0

    Show the following limit will converge to delta(y-b), lim 1/|a| f_x((y-b)/a)=delta(y-b) a-->0
  33. S

    Chemistry Need help with this threshold limit value (TLV) chemistry problem

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  34. S

    Finding the limit of a multivariable function

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  35. Adgorn

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  36. V

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  37. DaTario

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  38. Eclair_de_XII

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    Instinct tells me to just plug in the number, say the limit is zero, and be done with it. But at the same time, while reading the statement from the "Relevant equations" section of this post, I cannot feel but feel some doubt as to whether or not this is the right approach. I mean, only the...
  39. atommo

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    Let's say you are rubbing a balloon on your hair to make it charged. If you then discharge the balloon and rub it on your hair again (and repeat this process numerous times). Would your hair run out of electrons so eventually you would be unable to charge the balloon, or would your hair gain...
  40. lfdahl

    MHB Limit of integral challenge of (e^(-x)cosx)/(1/n+nx^2)

    Find \[\lim_{n\rightarrow \infty}\int_{0}^{\infty}\frac{e^{-x}\cos x}{\frac{1}{n}+nx^2}dx.\]
  41. Beelzedad

    I Are Extra Conditions Affecting the Limit Definition?

    This question consists of two parts: preliminary and the main question. Reading only the main question may be enough to get my point, but if you want details please have a look at the preliminary. PRELIMINARY: Let potential due to a small volume ##\delta## at a point ##(1,2,3)## inside it be...
  42. A

    I Let's talk about the classical limit of QM

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  43. B

    MHB TFAE proof involving limit and convergent sequence

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  44. B

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  45. Physics lover

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  46. E

    I Showing that a multivariable limit does not exist

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  47. Mcp

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  48. G

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  49. Physics lover

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  50. Physics lover

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