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. S

    I Proof that limit of difference of infinite limits is indeterminate

    (NOTE: I have had a few similar postings lately on this subject, but they were much broader in scope, so I am posting only for this particular case; everything else has been figured out.) If given that limx -> a f( x ) = +∞ limx -> a g( x ) = +∞ what is the epsilon-delta formulation for...
  2. MathematicalPhysicist

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

    What is exactly the definition of symmetric limit? It's the first place in the book that I see this notation, and they don't even define what it means. How does it a differ from a simple limit or asymptotic limit? I found a few hits in google, but it doesn't seem to help...
  3. M

    Some hypothetical limits for the Star Wars universe?

    I’ve read many Legends and Canon Star Wars books and I always take away stuff on their limits of technology and science. Over the years; here are some things they said science can’t do. 1.) Cybernetic liver- In Lost Stars, it was said Ciena’s liver could not be replaced as it was one of the...
  4. G

    I Finding an infinitesimal limit function

    I have the function: ##\sqrt{\left(\frac{x}{h}+1\right)^{2}+\left(\frac{y}{h}\right)^{2}}-\sqrt{\left(\frac{x}{h}\right)^{2}+\left(\frac{y}{h}\right)^{2}}## I would like to find an analytical solution, the equivalent function, in the limit of h approaching zero.Additional info which might be...
  5. PeterDonis

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

    A 2019 paper by Gorecki et al. derives an uncertainty principle limit that is larger than the conventional Heisenberg limit by a factor of ##\pi##: https://arxiv.org/abs/1907.05428 I'm wondering if any QM experts have seen this and what your thoughts are.
  6. bagasme

    B Rationale Behind t-Substitution for Evaluating Limits?

    Hello all, Given following limits: ##\lim_{x \rightarrow 1} {\frac {\sqrt x -1} {x^2 - 1}}## ##\lim_{x \rightarrow 1} {\frac {\sqrt {x+1} - 2} {x - 3}}## ##\lim_{x \rightarrow 1} {\frac {\sqrt[3] x - \sqrt[4] x} {\sqrt[6] x - \sqrt x}}## Those limits can be evaluated by letting ##x = t^2##...
  7. Witcher

    Finding Limits with DeltaX: An Essential Tool for Calculus

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  8. Merlin3189

    Is Double Dime Really Code for a 55mph Speed Limit?

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  9. M

    MHB Check existence of limit with definition

    Hey! :o I want to check the existence of the limit $\lim_{x\to 0}\frac{x}{x} $ using the definition. For that do we use the epsilon delta definition? If yes, I have done the following: Let $\epsilon>0$. We want to show that there is a $\delta>0$ s.t. if $0<|x-0|<\delta$ then...
  10. Z

    On/Off Ratio Limits: Practical Lower Limit for Logic Functionality

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  11. T

    I Dispersive approximation (limit) in the Jaynes-Cummings Model

    I wanted to know what is understood as the dispersive approximation (or limit) in the context of the Jaynes-Cummings model for one mode of the field.
  12. Wi_N

    This Limit problem seems too simple....

    $$\lim_{x\to\infty} (x^3+x^2 +\frac{x}{2})-x^3\sqrt{(1-\frac{1}{x^6})} = \lim_{x\to\infty} x^3+x^2+ \frac{x}{2} -x^3=\lim_{x\to\infty} x^2 + \frac{x}{2} = \infty. $$ is this it?
  13. 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##?
  14. Y

    MHB Limit involving a hyperbolic function

    Hello all, I am trying to solve a limit: \[\lim_{x\rightarrow 0}\frac{sinh (x)}{x}\] I found many suggestions online, from complex numbers to Taylor approximations. Finally I found a reasonable solution, but one move there doesn't make sense to me. I am attaching a picture: I have marked...
  15. Y

    MHB Limit of integer part function using Sandwich rule

    Hello everyone, I want to calculate the following limits: \[\lim_{x\rightarrow \infty }\frac{[x\cdot a]}{x}\] using the sandwich rule, where [xa] is the integer part function defined here: Integer Part -- from Wolfram MathWorld I am not sure how to approach this. Any assistance will be...
  16. Addez123

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

    I'm not sure if the way I solve these limits is correct, so let me know if I'm doing something wrong. $$\lim_{x^2+y^2 \rightarrow +\infty} {\frac {xy} {x^2+y^2}}$$ $$r = x^2+y^2$$ $$\lim_{r \rightarrow +\infty} {\frac {r\cdot cos(v) \cdot r \cdot sin(v)} r}$$ $$\lim_{r \rightarrow +\infty}...
  17. J

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

    Evaluation of $\displaystyle \lim_{n\rightarrow \infty}e^{-n}\sum^{n}_{k=0}\frac{n^k}{k!}$
  18. crises

    A Classical limit of the propagator

    I am currently starting with my first qft lectures and i am trying to see for the free particle that the propagator $$ <x_i | e^{-i\frac{p}{2m} T|x_f}>$$ will equal to one if x_f = 1, x_i=0 m=1 u=1 p=1, T=1 and $$\hbar \rightarrow 0$$ or 0 otherwise. I understand that this limit will result in...
  19. J

    MHB How to fully solve this limit evaluation using integration?

    Evaluation of \displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=1}\bigg(\frac{k}{n^2}\bigg)^{\frac{k}{n^2}+1}
  20. JTorn

    Why doesn't a real Infinite Redshift Limit occur at R+ for Kerr BHs?

    As I have studied before, I found that Infinite Red Shift occurs where gtt = 0 but this exercise says that on Kerr's Black Hole it doesn't really work like that. Right now I'm blocked because I didn't find anything on the internet about it so I don't know how to show this phenomenon. Any help...
  21. F

    Calculating the "mean values" in the thermodynamic limit

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  22. Amartansh

    Rayleigh limit in inverse scattering imaging

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  23. Wi_N

    Exploring the Validity of x=-2 as an Asymptote

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  24. N

    How Do You Solve a Limit as x Approaches Negative Infinity?

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  25. Adesh

    Calculus What are some books for learning the techniques of Calculus?

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  26. T

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  27. snatchingthepi

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  28. H

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  29. V

    Finding the limit using a trig identity

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  30. S

    Can Electrons Carry Infinite Energy?

    Are electrons limited to how much energy they can carry(if that term can be used)?
  31. 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
  32. S

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

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  33. S

    Finding the limit of a multivariable function

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  34. Adgorn

    Limit of the remainder of Taylor polynomial of composite functions

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  35. V

    Finding Maximum Delta for a Limit Involving a Quadratic Function

    Consider limx→3x^2=9. Find a maximum value of δ such that: |x2 - 9|<0.009 if |x-3|<δ I just learned how to do this today and I am quite comfortable doing this if the function is linear, however now I am struggling with working with quadratic functions. So far this is what I have come up with...
  36. DaTario

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  37. 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...
  38. atommo

    Limit to generating static electricity via contact?

    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...
  39. lfdahl

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

    Find \[\lim_{n\rightarrow \infty}\int_{0}^{\infty}\frac{e^{-x}\cos x}{\frac{1}{n}+nx^2}dx.\]
  40. 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...
  41. A

    I Let's talk about the classical limit of QM

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  42. B

    MHB TFAE proof involving limit and convergent sequence

    Let A ⊆ R, let f : A → R, and suppose that (a,∞) ⊆ A for some a ∈ R. Then the following statements are equivalent: i) limx→∞ f(x) = L ii) For every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn)) converges to L. Not even sure how to begin this one, other than the fact...
  43. B

    MHB Proving a limit to infinity using epsilon-delta

    lim 2x + 3 = ∞. x→∞ Pretty intuitive when considering the graph of the function. But how would I show this using the epsilon-delta definition?Thanks!
  44. Physics lover

    Simplified Limit Calculation for (1-e^(1-x/(1+x))x)/(1/x)

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  45. E

    I Showing that a multivariable limit does not exist

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  46. Mcp

    B A function's derivative being not defined for some X but having a limit

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  47. G

    B Is the case for a Universal Speed Limit experimental or theoretical?

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  48. Physics lover

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  49. Physics lover

    A trignometric limit going to infinity

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  50. M

    MHB Markov Process Limit: Calculating $u_k$ as a Function of $a,b$

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