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
If I am asked to show that the tt-component of the Einstein equation for the static metric
##ds^2 = (1-2\phi(r)) dt^2 - (1+2\phi(r)) dr^2 - r^2(d\theta^2 + sin^2(\theta) d\phi^2)##, where ##|\phi(r)| \ll1## reduces to the Newton's equation, what exactly am I supposed to prove?
Hi everybody! I have another question about integrability, especially about the limit comparison test. The script my teacher wrote states:
(roughly translated from German)
Limit test: Let -∞ < a < b ≤ ∞ and the functions f: [a,b) → [0,∞) and f: [a,b) → (0,∞) be proper integrable for any c ∈...
Homework Statement
##f(x)=12x^2-5##
The value of ##\lim_{h\rightarrow 0}\frac{f(x+2h)-f(x-3h)}{6h}## is ...
A. 8x
B. 10x
C. 12x
D.18x
E. 24x
Homework Equations
##f'(x) = \lim_{h->0}\frac{f(x+h)-f(x))}{h}##
The Attempt at a Solution
[/B]
Looking at the problem question, it seems that it's...
For the function given below find a formula for the Riemann sum obtained by dividing the interval [1,5] into n equal subintervals and using the right-hand endpoint for each c subscript k. Then take a limit of thissum as n-> infinite to calculate the area under the curve over [1,5].
Below you...
I've calculated the eigenstates of the Hubbard Hamiltonian for two fermions.
The ground state is (U2 - (U2 + 16t2)1/2)/2
For U = infty, I get 0.
For U >> t, I should get the exchange energy J = -4t2/U
How do I get from the ground state equation to J?
I am trying to explain to someone the formal notion of a limit of a function, however it has made me realize that I might have some faults in my own understanding. I will write down how I understand the subject and would very much appreciate if someone(s) can point out any...
Hi I was wondering how you get this when taking the limit of T going to 0
From this expression of S:
Please help I don't see how ln infinity goes to uB/KbT (used u to represent the greek letter. And how does the other expression of sinh and cosh approach 1?
... literature review that we are thinking of attempting to publish?
The review will be around 20,000 words and currently I have around 16000 words and 230 references, which seems quite a lot, is there a limit to how many references are present in a published review?
This is not a thesis, but...
I have heard that identical distinguishable classical particles having different ''statistics''.It is the limit of quantum case.Then we mix many parts(cells) of identical gases, the total entropy increases.I can not derive this limit from quantum particles to classical particles(please help...
Sunlight hits our planet, for example, and reflects light outward back into space. Hence why photos can be taken of Earth from outer-space. But if we disregard technological limits to optics, etc., then in theory how much information does this reflected light contain? Is it rich enough, for...
Homework Statement
I want to find conditions over A,B,C,D to observe a stable limit cycle in the following system:
\frac{dx}{dt} = x \; (A-B y) \hspace{1cm} \frac{dy}{dt} = -y \; (C-D x)
Homework Equations
Setting dx/dt=0 and dy/dt=0 you can find that (C/D , A/B) is a fixpoint.
The Attempt...
Hi All, I am desperate to understand a calculation presented in a paper by Sethna, "Elastic theory has zero radius of convergence", freely available online
$$ lim_{\epsilon \to +0}Z(-P+i\epsilon) = lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp \{...
Homework Statement
Let (an)n≥1 be a sequence with a1≥0 and an+1=an(an+4), n≥1. Compute limn→∞ (an)(1/(2n)).
Homework Equations
a1≥0
an+1=an(an+4), n≥1
L = limn→∞ (an)(1/(2n))
The Attempt at a Solution
Firstly, I had tried to see if an can be expressed only in terms of a1, but I couldn't get...
Originally from the statistics forum but am told this is more of a calculus question.
I flip 10 coins, if any of the coins land on tails, all of the coins split into 10 new coins and I flip them all again. I keep doing this until a round where every single coin lands on heads. Can I expect to...
Homework Statement
A wooden cube with the length of 10 cm is and a density of 700 kg/m^3 is floating on water
A)Find the submerged parts length of the cube
B) find the maximum added mass to the block before it becomes totally submerged
Homework Equations
FB=density*volume*gravity
FB=Fg...
A waiter believes the distribution of his tips has a model that is slightly skewed to the left, with a mean of $\$8.20$ and a standard deviation of $\$5.60$. He usually waits on about 60 parties over a weekend of work. a) Estimate the probability that he will earn at least $\$600$. b) How...
I've attached the equation for the transmission coefficient of a particle going through a potential barrier and E < V. I was simply wondering in the limit V --> E, why does T --> 0 (i.e. the V-E term --> 0 and thus the denominator would approach infinity, making T --> 0)? Shouldn't it be...
Homework Statement
If f(a) = 0 and f'(a) = 6 find lim h -> 0 (f(a+h)/2h).
Homework Equations
lim h ->0 (f(a+h)-f(a))/h
The Attempt at a Solution
I found the ratio between the two equations.
(f(a+h)-f(a))/h / (f(a+h)/2h)
I found this to be 2. Is this step possible or can you not take the ratio...
I know that the limit for the spherical bessel function of the first kind when $x<<1$ is:
j_{n}(x<<1)=\frac{x^n}{(2n+1)!}
I can see this from the formula for $j_{n}(x)$ (taken from wolfram's webpage):
j_{n}(x)=2^{n}x^{n}\sum_{k=0}^{\infty}\frac{(-1)^{n}(k+n)!}{k!(2k+2n+1)!}x^{2k}
and...
Piran et al., "Cosmic explosions, life in the Universe and the Cosmological Constant," http://arxiv.org/abs/1508.01034
I thought this was interesting. If I'm understanding correctly, the idea is that satellite galaxies such as the Magellanic Clouds have low metallicity, which causes them to...
Hi everyone,
So we were writting our math test today and I am not completely sure about one concept.
For the sake of simplicity let's say that
f(x)=x2
and let's say we were asked to find,
lim f(x) as x--->infinity = ?
is the correct answer here undefined or infinity.
Thanks for the help
Why does the following Lagrangian not have the correct non-relativistic limit? It is correct except for the derivative of proper time with respect time. But that factor goes to 1 so why is the expression wrong?
## L = -(\frac{1}{2}mu^{\mu}u_{\mu} + qu^{\mu}A_{\mu})\frac{d\tau}{dt} ##
Homework Statement
Prove that
lim (x,y,z)→(0,0,0) 2xz/(x²+y²+z²) = 0
Homework Equations
My teacher wants me to show this using epsilon delta, so
0<√(x²+y²+z²)<∂ ⇒ |f(x,y,z) - 0| < ε
The Attempt at a Solution
The limit does not exist apparently.. when you approach the limit along different...
When we integrate $f'(x)$ we get $f(x)$ and say we integrate from x = a to b
in the output we write f(x) within square brackets and limit on the right.
how do I write in latex
thanks in advance.
Homework Statement
For what value of the constants a and b such that the following limit exists?
lim {(ax+|x+1|)|x+b-2|}/|x+1|
x->-1 help me ,thx
Homework EquationsThe Attempt at a Solution
first, I know that I should cancel the absolute value at denominator of x+1. but i don't how to...
$$\lim_{{x}\to{0}}\frac{\cos\left({3x}\right)-1}{{x}^{2}}$$
$$\frac{f'}{g'}
=-\frac{3\sin\left({3x}\right)}{2x}
=-\frac{9}{2}\cdot\frac{\sin\left({3x}\right)}{3x}$$
$x\to 0$ is $-\frac{9}{2}$
Just seeing if this is correct or better way to do it
Homework Statement
Find \lim_{x \to 0}\frac{ln(1+x^2)}{1-cos(x)} by using series representations. Check using L'Hospitals rule.
Homework Equations
Taylor polynomial at x=0: \sum_{k=0}^{\infty}\frac{f^{k}(0)}{k!}(x)^{k} = f(0) + f'(0)(x) + f''(0)x^{2} +...
The Attempt at a Solution
Using...
Homework Statement
Assume f:(a,b)→ℝ is differentiable on (a,b) and that |f'(x)| < 1 for all x in (a,b). Let an
be a sequence in (a,b) so that an→a. Show that the limit as n goes to infinity of f(an) exists.
Homework Equations
We've learned about the mean value theorem, and all of that fun...
Homework Statement
Hi, I'm currently studding a module on PLC and have a question on "what will happen if you do this ...".
Homework Equations
I don't have a problem with explaining operation of each rungs of the ladder diagram, I need some help on explaining the function of the reverse relay...
Hi, I'm currently studding a module on PLC and have a question on "what will happen if you do this ...". As I don't have a problem with explaining operation of each rungs of the ladder diagram I need some help on explaining the function of the reverse relay and travel limit switches. Can anyone...
In the system of units where c=1, the Lorentz transformations are as follows:
## x'=\gamma (x-vt) \\ t'=\gamma (t-vx) ##
In the limit ## v \ll 1 ##, we have ## \gamma \approx 1+\frac 1 2 v^2 ##, so we have, in this limit:
## x' \approx (1+\frac 1 2 v^2)(x-vt)=x-vt+\frac 1 2 v^2 x-\frac 1 2...
I don't understand why the limit of x1/loga(x) as x approaches infinity is a, where a can be any constant for the base.
Why isn't it 1? The base (x), approaches infinity, while the exponent approaches 0 (1/infinity), so it should be (infinity)0 = 1.
what value of the constants a and b if the following limit exists
lim (ax + |x + 1|)|x + b − 2| |x + 1|
x→−1
|x|= x for x≥ 0 and |x|= -x for x<0
|x+1|= x+1 for x≥ -1 and |x+1|= -(x+1) for x<-1
I don't know how to determine |x + b − 2| is positive or negative.
i know that if limit...
Lim as x approaches 4 of 1/x = 1/4
Given epsilon > 0, come up with a delta, d?Limits have been introduced. So far my instructor has had us make tables to see what value x was approaching. Although I don't understand exactly how limits are EVALUATED (different from looking at a chart & saying...
Homework Statement
The problem wants me to find the limit below using series expansion.
##\lim_{x \to 0}(\frac{1}{x^2}\cdot \frac{\cos x}{(\sin x)^2})##
Homework EquationsThe Attempt at a Solution
(1) For startes I'll group the two fractions inside the limit together
##\lim_{x...
Homework Statement
I have to prove that the limit as n ⇒ ∞ of: n(a1/n-1) = log(a) -> For every a >0
Homework Equations
I have no idea what to use
The Attempt at a Solution
This was an exam question i left it blank, because i had (and have) no idea on where to even start, I've tried using...
I have the following limit to calculate under the assumption that ##\Re(x+y)>1##:
Limit[Integrate[
1/((1 + t^2)^n*(1 + I*t)^x (1 - I*t)^y), {t, -Infinity, Infinity}],
n -> Infinity]
I want to add the above assumption for integral, how to do it? does it even converge?
Thanks.
In a homework problem I had to find the limit as x goes to 0 of the function: sin(7x)/[x+tan(9x)]
Substituting sin(9x)/cos(9x) in for tan(9x) then dividing the top and bottom by x and finding the limit supposedly yields 7/1+(9)(1), giving an answer of 7/10.
What I don't get is why the limit as...
In my book, applied analysis by john hunter it gives me a strange way of stating an infinite sum that I'm still trying to understand because in my calculus books it was never described this way.
It says:
We can use the definition of the convergence of a sequence to define the sum of an...
What is the ##\lim_{x \to \infty} x^2##?
What I get is:
##\lim_{x \to \infty} x^2##
##= \lim_{x \to \infty} \frac{\frac{1}{x^2}}{\frac{1}{x^2}} x^2##
##= \lim_{x \to \infty} \frac{\frac{x^2}{x^2}}{\frac{1}{x^2}}##
##= \lim_{x \to \infty} \frac{1}{\frac{1}{x^2}}##
##=...
Homework Statement
Use the epsilon delta definition to show that lim(x,y) -> (0,0) (x*y^3)/(x^2 + 2y^2) = 0
Homework Equations
sqrt(x^2) = |x| <= sqrt(x^2+y^2) ==> |x|/sqrt(x^2+y^2) <= 1 ==> |x|/(x^2+2y^2)?
The Attempt at a Solution
This limit is true IFF for all values of epsilon > 0, there...
Homework Statement
given a geometric sequence sin(x),sin(2x), . . .
c) find for which values of x∈(0,π) this sequence converges and calculate its limit
Homework Equations
|q|<1 or -1<q<1The Attempt at a Solution
Ok so in part a) and b) i calculated the quotient and found out that...