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. Lotto

    How to calculate primitive functions on maximal intervals for periodic functions?

    Since ##f(x)## is continuous in ##\mathbb R##, it has a primitive function in ##\mathbb R## as well, so we have to define ##F(x)## also for points ## \frac{\pi}{2}+k\pi##. ##\lim_{x \to \left(\frac{\pi}{2}+k\pi \right)^-} F(x) =\frac{\pi}{2}+k\pi -\frac{\pi}{2\sqrt 2}+C_k ## ##\lim_{x \to...
  2. chwala

    I Understanding the ##ε## as used in limits of sequences

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  3. T

    Prove Continuity From Precise Definition of Limit

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  4. S

    Prove that the limit as n->infinity of n^n/n! is infinity

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  5. Agent Smith

    B Central Limit Theorem: How does sample size affect the sampling distribution?

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  6. chwala

    Evaluate the limit of the given problem

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  7. S

    Questions about information limits in the Universe

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  8. S

    Finding limit using integration

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  9. A

    Is there a limit on how much energy a photon might have in a FOR?

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  10. M

    Proving limit of rational function

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  11. T

    B Why 186,282?

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  12. L

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  13. L

    Help calculating this limit please

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  14. C

    B Question about the fundamental theorem of calculus

    Hello everyone, I've been brushing up on some calculus and had some new questions come to mind. I notice that most proofs of the fundamental theorem of calculus (the one stating the derivative of the accumulation function of f is equal to f itself) only use a limit where the derivative is...
  15. A

    Evaluate the limit of this harmonic series as n tends to infinity

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  16. L

    Calculate the limit cos(x)/sin(x) when x approaches 0

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  17. L

    Show that the limit (1+z/n)^n=e^z holds

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  18. MatinSAR

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  19. Weightlifting

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  20. S

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  21. S

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  22. Lotto

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  23. MichPod

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  24. Euge

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  25. Infrared

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  26. Euge

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  27. E

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  28. Mohmmad Maaitah

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    I'm talking about the x^(-4/3) how does it equal 0 when x approch infinite?? so I can use L'Hopital's Rule
  29. kornelthefirst

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  30. M

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  31. Lotto

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    I think that the work is meant to be work done for instance in power stations. Or is it similar to work I do on a body when I lift it for example? But how can we then do that work on our Earth? I just need to understand the task, otherwise I want to solve it myself. The problem involves...
  32. M

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  33. ananonanunes

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    This is what I did: $$\lim_ {(x,y) \rightarrow (1,0)} {\frac {g(x)(x-1)^2y}{2(x-1)^4+y^2}}=\lim_ {(x,y) \rightarrow (1,0)} {g(x)y\frac {(x-1)^2}{2(x-1)^4+y^2}}$$ I know that ##\lim_ {(x,y) \rightarrow (1,0)} {g(x)y}=0## and that ##\frac {(x-1)^2}{2(x-1)^4+y^2}## is limited because ##0\leq...
  34. C

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  35. M

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    For this problem, The solution is, However, when I tried finding the domain myself: ## { x | x - 1 ≥ \sqrt{5}} ## (Sorry, for some reason the brackets are not here) ##{ x | x - 1 ≥ -\sqrt{5}} ## and ## { x | x - 1 ≥ \sqrt{5}}## ##{x | x ≥ 1 -\sqrt{5} }## and ## { x | x ≥ \sqrt{5} + 1}##...
  36. B

    I Limit as a function, not a value

    Is it possible for a limit of a range of functions to return a function? Example: f(z)= limit (as p approaches 0) (xp-1)/p.
  37. O

    I Limit of quantum mechanics as h -> 0

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  38. M

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  39. mcastillo356

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  40. G

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  41. Euge

    POTW Limit of Complex Sums: Find $$\lim_{n\to \infty}$$

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  42. PeterDonis

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  43. V

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  44. C

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  45. L

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  46. murshid_islam

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  47. Rikudo

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  48. murshid_islam

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  49. X

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