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. Dethrone

    MHB What is the limit of x^n/n as n approaches infinity?

    Prove that $\lim_{{n}\to{\infty}}\frac{x^n}{n!}=0$.
  2. Dethrone

    MHB Prove $\lim_{{n}\to{\infty}}(3^n+4^n)^{1/n}=4$

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  3. Feodalherren

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  4. N

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  5. S

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  6. 22990atinesh

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  7. Philethan

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  8. L

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  9. K

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  10. Nathanael

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  11. J

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  12. L

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  13. Y

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  14. M

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  15. C

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  16. ubergewehr273

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  17. M

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  18. Euclid Areti

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  19. M

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  20. Y

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  21. Demystifier

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  22. M

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  23. Randall

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  24. T

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  25. Y

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  26. Randall

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  27. Randall

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  28. K

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    Can someone here help me establish that $\lim\limits_{t \rightarrow \infty} S(t) = \Lambda/\mu$, given that: $\frac{dS}{dt}=\Lambda-\mu S-\beta \frac{S}{N}(H+C+C_1+C_2)-\tau \frac{S}{N}(T+C)$ $N(t)=S(t)+H(t)+C(t)+C_1(t)+C_2(t)$ $\Lambda,\beta,\tau > 0$ $\text{As} \ \ t \rightarrow \infty, \...
  29. Warpspeed13

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  30. R

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    This is a basic formula but can't find any proof of it. If anyone can explain or give a link showing proof it would be helpful. ## \lim_{x\rightarrow a} f(x)^{g(x)} ## = ##e^{\lim_{x\rightarrow a} g(x)[f(x)-1]}## What is the proof for it?
  31. andyrk

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  32. G

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  33. C

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  34. Maged Saeed

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  35. wolfspirit

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  36. H

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    Hello, I was just wondering, we have what could be called the indefinite derivative in the form of d/dx x^2=2x & evaluating at a particular x to get the definite derivative at that x. But with derivation, we can algebraically manipulate the limit definition of a derivative to actually evaluate...
  37. J

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  38. J

    MHB Left hand and right hand limit at infinity

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  39. Y

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  40. A

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  41. T

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  42. F

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  43. M

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  44. A

    MHB Assume the Limit Exists: Proving an Impossibility

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  45. A

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  46. M

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  47. A

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  49. ironman

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  50. G

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