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. chwala

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

    I will create my own example on this- Phew atleast this concepts are becoming clearer ; your indulgence is welcome. Let me have a sequence given as, ##Un = \dfrac {7n-1}{9n+2}## ##Lim_{n→∞} \left[\dfrac {7n-1}{9n+2}\right] = \dfrac {7}{9} ## Now, ##\left[ \dfrac {7n-1}{9n+2} - \dfrac {7}{9}...
  2. T

    Prove Continuity From Precise Definition of Limit

    I attached my attemp at the solution. I am trying to start with continuity at 0 and end up with limit of f(x) equals f(c) as x goes to c. Could someone take a look at the attached image and let me know if I am on the right track or where I went astray Sorry picture is rotated I tried but...
  3. S

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

    I am self-studying Boas and this is a problem from Ch. 1.2. I have developed what I believe is an answer, but I'm not sure it's adequate. The general approach is to show that for all values of n > 1, n^n grows faster than n!, and therefore that (n^n)/n! approaches infinity as n approaches...
  4. Agent Smith

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

    In this course I took it says that the larger the sample size the more likely is the sampling distribution (of the sample means, guessing here) to be normal. This they say is The Central Limit Theorem. How does this work? How does someone taking a large sample affect the sampling distribution...
  5. chwala

    Evaluate the limit of the given problem

    Ok, to my question; was the step highlighted necessary? The working steps to solution are clear. Cheers.
  6. S

    Questions about information limits in the Universe

    For a hammer all problems are nails, and for a signal processing animal like me, it is all about information :cool: The laws of mechanics limit the info needed to describe/define motion. The limited number of finite sized stable elements, built using smaller number of particles, limit the...
  7. S

    Finding limit using integration

    I want to ask why the answer is not zero. If n approaches infinity, it means each term will approach zero so why the answer is not zero? Thanks
  8. A

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

    So there was this question: The first option seems to be the only correct answer. $$\lambda_e=\dfrac{h}{\sqrt{2m(KE)}}$$. The answer would be correct if ##KE \approx eV## The option mentions that ##eV>>\phi## so ##\phi## can be ignored. But I don't think that necessarily means that the...
  9. M

    Proving limit of rational function

    For this problem, The solution is, However, I'm confused how ##0 < | x - 1|< 1## (Putting a bound on ##| x- 1|##) implies that ##1 < |x+1| < 3##. Does someone please know how? My proof is, ##0 < | x - 1|< 1## ##|2| < | x - 1| + |2| < |2| + 1## ##2 < |x - 1| + |2| < 3## Then take absolute...
  10. T

    B Why 186,282?

    Is there an explanation for why the speed of light tops out at 186,282 miles per second? Of course that number depends on our definition of miles and seconds. If a mile was 3000 feet then c would be a different number. But whatever speed it is…. Why that speed? In other words… there is...
  11. L

    I It's still not clear to me what's the limit of light propagation

    Hi, I've been asking questions about light here for years, and I still don't understand the limit of propagation of light, does anyone have advanced on this field? I really would like someone to explain me how it's possible for light to propagate forever, since it's probably emitted in perfect...
  12. L

    Help calculating this limit please

    Hi, I have problems with task b, more precisely with the calculation of the limit value: By the way, I got the following for task a ##f^{(n)}(x)=(-1)^{n+1} \frac{(n-1)}{x^n}## Unfortunately, I have no idea how to calculate the limit value for the remainder element, since ##n## appears in...
  13. 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...
  14. A

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

    To use the formula above, I have to prove that $$\lim_{n\rightarrow \infty}f(x)=\lim_{n\rightarrow \infty}\left(\frac{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+........\frac{1}{n}}{n^2}\right)=1$$ To prove so, I tried using L'Hopital's Rule: $$\lim_{n\rightarrow \infty}f(x)=\lim_{n\rightarrow...
  15. L

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

    Hi, I need to check whether the limit of the following function exists or not I have now proceeded as follows to look at the right-sided and left-sided limit i.e. ##\displaystyle{\lim_{x \to 0^{+}}}## and ##\displaystyle{\lim_{x \to 0^{-}}}## To do this, I drew up a list in which I move from...
  16. L

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

    Hi, I have problems proving task d I then started with task c and rewrote it as follows ##\lim_{n\to\infty}\sum\limits_{k=0}^{N}\Bigl( \frac{z^k}{k!} - \binom{n}{k} \frac{z^k}{n^k} \Bigr)=0 \quad \rightarrow \quad \lim_{n\to\infty}\sum\limits_{k=0}^{N} \frac{z^k}{k!} =...
  17. MatinSAR

    Find ##b-a## which satisfies following limit

    ## \lim_{x \rightarrow 1} {\frac {x-2} {x^3+ax+b}} = -\infty## The limit is equal to ##\frac {-1} {1+a+b}## . so I can say that ## a+b = -1 ##. But I cannot find another equation to find both ##b-a##.
  18. Weightlifting

    Finding the direction of infinite limits

    Since ##\left|3x+2\right|=0\rightarrow\ x=-\frac{2}{3}##, we know the vertical asymptote is at ##x=-\frac{2}{3}##. Looking at the limit at that point, and also looking at the left- and right-sided limit, I cannot simplify it any further...
  19. S

    Limit and Integration of ##f_n (x)##

    My attempt: (a) I don't think I completely understand the question. By "evaluate ##\lim_{n\to \infty f_n (x)}##", does the question ask in numerical value or in terms of ##x##? As ##x## approaches 1 or -1, the value of ##f_n (x)## approaches zero. As ##x## approaches zero, the value of ##f_n...
  20. S

    Proof related to derivative and Big O notation

    My attempt: Since ##\frac{f(a+h)-f(a-h)}{2h}-f'(a)=O(h^2)## as ##h \to 0##, then: $$\lim_{h \to 0} \frac{\frac{f(a+h)-f(a-h)}{2h}-f'(a)}{h^2} < \infty$$ So $$\lim_{h \to 0} \frac{\frac{f(a+h)-f(a-h)}{2h}-f'(a)}{h^2} = \lim_{h \to 0} \frac{f'(a)-f'(a)}{h^2}=0 < \infty$$ Because the value of...
  21. Lotto

    What is density of dark matter as a function of distance from the galactic core?

    This problem builds on my previous post, where we calculated that core's mass is ##M_1=\frac{{v_0}^2r_1}{G}##. So if we consider mass of dark matter dependent on distance ##r## to be ##M_2(r)##, we can calculated it from ##G\frac{(M_2(r)+M_1)m}{r^2}=m\frac{{v_0}^2}{r}.## So...
  22. MichPod

    Does limit 1/x at zero equal infinity? How it is accepted in High School now?

    Hello. Per what I was taught in my youth, ##\lim_{x \to 0}\frac{1}{x}=\infty## Is it in agreement with how the calculus is taught today in the High Schools and Universities of US/Canada specifically? Per what my son says, that limit should be considered as undefined because ##\lim_{x \to...
  23. Euge

    POTW Power Sums Limits: Evaluating $\lim_{n\to \infty}$

    If ##n## and ##k## are positive integers, let ##S_k(n)## be the sum of ##k##-th powers of the first ##n## natural numbers, i.e., $$S_k(n) = 1^k + 2^k + \cdots + n^k$$ Evaluate the limits $$\lim_{n\to \infty} \frac{S_k(n)}{n^k}$$ and $$\lim_{n\to\infty} \left(\frac{S_k(n)}{n^k} -...
  24. Infrared

    Challenge Math Challenge - July 2023

    Welcome to this month's math challenge thread! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Have fun! 1. (solved by @AndreasC) I start watching a...
  25. Euge

    POTW Find Limit of $$\frac{x}{e} - \left(\frac{x}{x+1}\right)^x$$ at Infinity

    Find the limit $$\lim_{x\to \infty} x\left[\frac{1}{e} - \left(\frac{x}{x+1}\right)^x\right]$$
  26. E

    Details regarding the high temperature limit of the partition function

    My main question here is about how we actually justify, hopefully fairly rigorously, the steps leading towards converting the sum to an integral. My work is below: If we consider the canonical ensemble then, after tracing over the corresponding exponential we get: $$Z = \sum_{n=0}^\infty...
  27. Mohmmad Maaitah

    L'Hopital's Rule case: How does x^(-4/3) equal 0 when x approches infinity?

    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
  28. kornelthefirst

    What Is the Next Step in Simplifying the Van der Pol Oscillator Equation?

    First i looked at the case of ## \epsilon = 0## and came to the conclusion, that this oscillator has a circular limit cycle in a phase space trajectory, when plotted with the axes x and ##\dot{x}##. I have found that ##x_0^p (t) = a_1 \cos(t)## which implies that all other Fourier- coefficients...
  29. M

    Limit problem involving two circles and a line

    For this problem, The limiting position of R is (4,0). However, I am trying to solve this problem using a method that is different to the solutions. So far I have got, ##C_1: (x - 1)^2 + y^2 = 1## ##C_2: x^2 + y^2 = r^2## To find the equation of PQ, ## P(0,r) ## and ##R(R,0) ## ## y =...
  30. Lotto

    B What is meant by "upper limit of work done on Earth"?

    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...
  31. M

    Using continuity to evaluate a limit of a composite function

    For this problem, The solution is, However, I tried to solve this problem using my Graphics Calculator instead of completing the square. I got the zeros of ##x^2 - 2x - 4## to be ##x_1 = 3.236## and ##x_2 = -1.236## Therefore ##x_1 ≥ 3.236## and ##x_2 ≥ -1.236## Since ##x_1 > x_2## then...
  32. ananonanunes

    Find limit of multi variable function

    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...
  33. C

    Limit on the edge of the domain

    What is the limit of the function as x goes to -5 (e.g. in the graph below) if the domain of the function is only defined on the closed interval [-5,5]? I realize that the right hand limit DOES exist and is equal to 3, but the left hand limit does not exist? So does that mean that the overall...
  34. M

    I Finding domain when using continuity to evaluate a limit

    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}##...
  35. 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.
  36. O

    I Limit of quantum mechanics as h -> 0

    Starting from the Heisenberg equation of motion, we have $$ih \frac{\partial p}{\partial t} = [p, H]$$ which simplifies to $$ih \frac{\partial p}{\partial t} = -ih\frac{\partial V}{\partial x}$$ but this just results in ## \frac{\partial p}{\partial t} = -ih\frac{\partial V}{\partial x}## and...
  37. M

    Limit of a rational function with a constant c

    For this problem, Did they get ## x## approaches one is equivalent to ##t## approaches zero because ##t ∝ (x)^{1/3} + 1##? Many thanks!
  38. mcastillo356

    I Express the limit as a definite integral

    Hi, PF, there goes the definition of General Riemann Sum, and later the exercise. Finally one doubt and my attempt: (i) General Riemann Sums Let ##P=\{x_0,x_1,x_2,\cdots,x_n\}##, where ##a=x_0<x_1<x_2<\cdots<x_n=b##, be a partition of ##[a,b]##, having norm ##||P||=\mbox{max}_{1<i<n}\Delta...
  39. G

    Calculate limit value with several variables

    Hi, I had to calculate the entropy in a task of a lattice gas and derive a formula for the pressure from it and got the following $$P=\frac{k_b T}{a_0}\Bigl[ \ln(\frac{L}{a_0}-N(n-1)-\ln(\frac{L}{a_0}-nN) \Bigr]$$ But now I am supposed to calculate the following limit $$\lim\limits_{a_0...
  40. Euge

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

    Let ##c## be a complex number with ##|c| \neq 1##. Find $$\lim_{n\to \infty} \frac{1}{n}\sum_{\ell = 1}^n \frac{\sin(e^{2\pi i \ell/n})}{1-ce^{-2\pi i \ell/n}}$$
  41. PeterDonis

    A Landau's Maximum Mass Limit Derivation in Shapiro & Teukolsky (1983)

    In Section 3.4 of Shapiro & Teukolsky (1983), a simple derivation, due to Landau, of the maximum mass limit for white dwarfs and neutron stars is given. I will briefly describe it here and then pose my question. The basic method is to derive an expression for the total energy (excluding rest...
  42. V

    Limit question to be done without using derivatives

    I am confused by this question. If I try applying the theorem under Relevant Equations then it seems to me that the theorem cannot be applied since the limit of the denominator is zero. This question needs to be done without using derivatives since it appears in the Limits chapter, which...
  43. C

    B Deep Space Speed Limit: What Prevents Exceeding Light Speed?

    This is probably a dumb question. I'm not a physicist and took basic physics a very long time ago. If an object was in deep space, a long way away from gravitational fields and was subjected to a constant 1g acceleration in a straight line what prevents it from eventually exceeding light speed...
  44. L

    I Limit of the product of these two functions

    If we have two functions ##f(x)## such that ##\lim_{x \to \infty}f(x)=0## and ##g(x)=\sin x## for which ##\lim_{x \to \infty}g(x)## does not exist. Can you send me the Theorem and book where it is clearly written that \lim_{x \to \infty}f(x)g(x)=0 I found that only for sequences, but it should...
  45. murshid_islam

    I Verifying Integration of ##\int_0^1 x^m \ln x \, \mathrm{d}x##

    I'm trying to compute ##\int_0^1 x^m \ln x \, \mathrm{d}x##. I'm wondering if the bit about the application of L'Hopital's rule was ok. Can anyone check? Letting ##u = \ln x## and ##\mathrm{d}v = x^m##, we have ##\mathrm{d}u = \frac{1}{x}\mathrm{d}x ## and ##v = \frac{x^{m+1}}{m+1}##...
  46. Rikudo

    Proving the result of the following limit

    Right now, I am trying to prove this : I tried to use this identity to solve it: Then, the limit will become ##\frac {x}{e-e}## However, the result is still ##\frac 0 0 ## Could you please give me hints to solve this problem?
  47. murshid_islam

    I Question about Limit: \lim_{x\rightarrow1} (x+1)

    \lim_{x \rightarrow 1} \frac{x^2 - 1}{x-1} For this, we first divide the numerator and denominator by (x-1) and we get \lim_{x \rightarrow 1} (x+1) Apparently, we can divide by (x-1) because x \neq 1, but then we plug in x = 1 and get 2 as the limit. Is x = 1 or x \neq 1? What exactly is...
  48. X

    Calculating Limits without L'Hopital: A Scientist's Perspective

    How can I calculate preferably without L'Hopital? Thanks.
  49. G

    I Geodesic in Weak Field Limit: Introducing Einstein's Relativity

    I'm reading《Introducing Einstein's Relativity_ A Deeper Understanding Ed 2》on page 180,it says: since we are interested in the Newtonian limit,we restrict our attention to the spatial part of the geodesic equation,i.e.when a=##\alpha####\quad ##,and we obtain,by using...
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