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. RJLiberator

    Multivariable limit problem with cos/cos

    Homework Statement Lim (x,y) --> (pi, 0) of (cos(x-y))/(cos(x+y)) Homework Equations The answer is 1 The Attempt at a Solution My answer is this: The function is continuous at the point in question, so we only need to plug in the values which result to be 1. My question here: I know this...
  2. Dethrone

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  3. Dethrone

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

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  4. Feodalherren

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  5. N

    Limit of e^-7x cos x: Using Squeeze Theorem for Calculus Homework

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  6. S

    Limit as x tends to infinity, without Laurent

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  7. 22990atinesh

    How to Solve a Limit Problem Involving Logarithms?

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  8. Philethan

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  9. L

    MHB Help With Limit: a=16, f=${x}^{.25}$

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  10. K

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  11. Nathanael

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  12. J

    Does e^{-ix} Have a Limit as x Approaches Infinity?

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  13. L

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  14. Y

    MHB Limit of a function when x goes to -infinity

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  15. M

    Limit of sin((pi*x*y)/4) at (x,y)--->(-1, 6): 1

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  16. C

    Proving Limit Laws: Epsilon/Delta Definition

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  17. ubergewehr273

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  18. M

    Help with Complex Limit: s, N → ∞

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  19. Euclid Areti

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  20. M

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  21. Y

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  22. Demystifier

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  23. M

    What Is the Long-Term Behavior of Radioactive Substance Mass Q(t)?

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  24. Randall

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  25. T

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  26. Y

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    The kinetic limited mass flux due to evaporation at the liquid–vapor interface surrounded by an air– vapor mixture is given as: mass flux = 2σ/(2-σ) *(M/2piRT)^0.5 * (Pv,eq -Pv) where Pv,eq is the saturated vapor pressure of liquid and Pv is the partial pressure of vapor in the gas phase very...
  27. Randall

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  28. Randall

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  29. K

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  30. Warpspeed13

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  31. R

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  32. andyrk

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  33. G

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  34. C

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  35. Maged Saeed

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  36. wolfspirit

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  37. 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...
  38. J

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  39. J

    MHB Left hand and right hand limit at infinity

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  40. Y

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  41. A

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  42. T

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  43. F

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  44. M

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  45. A

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  46. A

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  47. M

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  48. A

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  49. V

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  50. ironman

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