What is Limit: Definition and 1000 Discussions

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

.

The limit superior of a sequence

x

n

{\displaystyle x_{n}}
is denoted by

lim sup

n

x

n

or

lim
¯

n

x

n

.

View More On Wikipedia.org
1. 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...

12. 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 Do not cite theorems that trivialize the problem you're solving. 3.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...
13. POTW Find Limit of $$\frac{x}{e} - \left(\frac{x}{x+1}\right)^x$$ at Infinity

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

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

Let ##c## be a complex number with ##|c| \neq 1##. Find $$\lim_{n\to \infty} \frac{1}{n}\sum_{\ell = 1}^n \frac{\sin(e^{2\pi i \ell/n})}{1-ce^{-2\pi i \ell/n}}$$
29. 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...
30. 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...
31. 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...
32. 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...
33. 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}##...
34. 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?
35. 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...
36. Calculating Limits without L'Hopital: A Scientist's Perspective

How can I calculate preferably without L'Hopital? Thanks.
37. 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...
38. Standard topology is coarser than lower limit topology?

Hello everyone, Our topology professor have introduced the standard topology of ##\mathbb{R}## as: $$\tau=\left\{u\subset\mathbb{R}:\forall x\in u\exists\delta>0\ s.t.\ \left(x-\delta,x+\delta\right)\subset u\right\},$$ and the lower limit topology as...
39. Determine whether limit is indeterminate or has a fixed value

Indeterminate forms are: ##\frac{0}{0}, \frac{\infty}{\infty} , \infty - \infty, 0 . \infty , 1^{\infty}, 0^{0}, \infty^{0}## My answer: 4, 9, 15, 17, 20 are inderterminate forms 1. always has a fixed finite value, which is zero 2. ##0^{-\infty}=\frac{1}{0^{\infty}}=\frac{1}{0}=\infty## so it...
40. A Limit of ##i^\frac{1}{n}## as ##n \to \infty##

##i^\frac{1}{n}## has n roots. If one is not careful, the limit as ##n \to \infty## is 1. Simple proof: ##i=e^\frac{\pi i}{2}## or ##i^\frac{1}{n}=e^\frac{\pi i}{2n} \to e^0=1##. This does not take into account the n roots, since ##i=e^{(\pi i)(2k+\frac{1}{2})}##.. Here ##\frac{k}{n} ## can...
41. Proving limit of f(x), f'(x) and f"(x) as x approaches infinity

I imagine ##f(x)## has horizontal asymptote at ##x=k##. Since the graph of ##f(x)## will be close to horizontal as ##x \rightarrow \infty##, the slope of the graph will be close to zero so ##\lim_{x \rightarrow \infty} f'(x) = \lim_{x \rightarrow \infty} f^{"} (x) = 0## But how to put it in...
42. B Is there a theoretical size limit for a planet?

Jupiter is huge. TrES-4 is 1.8 times the size. How big can planets actually get? is there a limiting factor? cheers.
43. Limit Definition of Derivative as n Approaches Infinity

##f'(x_0)## is defined as: $$f'(x_0)=\lim_{h \rightarrow 0} \frac{f(x_0+h)-f(x_0)}{h}$$ or $$f'(x_0)=\lim_{x \rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}$$ I can imagine that as ##n \rightarrow \infty## the value of ##f(b_n)## and ##f(a_n)## will approach ##f(x_0)## so the value of the limit will...
44. Find the values of a and b in a limit

I know $$\lim_{h\rightarrow 0} af(h)+bf(2h)−f(0)=0$$ $$a+b=1$$ But I don't know how to find the second equation involving a and b. I imagine I need to somehow obtain ##h## in numerator so I can cross out with ##h## in denominator but I don't have idea how to get ##h## in the numerator. Thanks
45. Prove: Limit Point of H ∪ K if p is Limit Point of H or K

Summary: Definition: If M is a set and p is a point, then p is a limit point of M if every open interval containing p contains a point of M different from p. Prove: that if H and K are sets and p is a limit point of H ∪ K,then p is a limit point of H or p is a limit point of K In this proof I...
46. A The distribution that has a certain distribution as its limit case

I have a probability distribution of the following form: $$\displaystyle f \left(t \right) \, = \, \frac{\lambda ~e^{-\frac{\lambda ~t }{k }}}{k }, \, 0 < t, \, 0< \lambda, \, k = 1, 2, 3, \dots$$ It seems that this distribution is a limiting case of another distribution. The question is what...
47. Solution to Differential Equation with Limit Boundary Condition

The original differential equation is: My solution is below, where C and D are constants. I have verified that it satisfies the original DE. When I apply the first boundary condition, I obtain that , but I'm unsure where to go from there to apply the second boundary condition. I know that I...

50. I Upper limit of relativistic spin?

Was curious at the upper limit for neutron stars, found this article stating one was found at around 700 / s https://www.newscientist.com/article/dn8576-fast-spinning-neutron-star-smashes-speed-limit/ did not see the size, the article is behind a paywall, but it would have taken a radius of...