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
Homework Statement
http://s14.postimg.org/an6f4t2ht/Untitled.png
Homework EquationsThe Attempt at a Solution
I'm not sure what they want me to do on the last part. I tried some googling and looking in my textbooks but I didn't find any examples.
It seems to me like the function goes to...
(sorry the thread title is wrong - can a mod please change it to "Limit of e^-7x cos x?")
1. Homework Statement
Find the following:
\lim_{x \rightarrow \infty} e^{-7x} \cos x
Homework Equations
I know that [ \lim_{x \rightarrow a} f(x)g(x) ] = [ \lim_{x \rightarrow a} f(x) ] \cdot [ \lim_{x...
Homework Statement
I want to find the following limit, ## \lim_{x \rightarrow \infty } x( \sqrt{ x^{2} +9} -x) ##, without using the Laurent series
Homework Equations
None.
The Attempt at a Solution
I used the Laurent Series to expand the square root, giving ## x((x+\frac{9}{2x})-x)##, then...
Suppose there is a limit
##\lim_{n \to \infty} \frac{n^{1.74}}{n \times (\log n)^9}##
Taking logs both on numerator and denominator
##=\lim_{n \to \infty} \frac{1.74 \times \log n}{\log n + 9 \log \log n}##
What can we say about the limit as n approaches ##\infty##
As I read in the James Stewart's Calculus 7th edition, he said:
My question is: Is f(x)\rightarrow 0 the same as f(x) = L?
For example,
f(x) = x^2
\displaystyle\lim_{x\rightarrow 5}f(x) = 25
I can say that f(x) = x^2 approaches 25 as x approaches 5.
Therefore, can I say that the...
Hello, I have this homework questions with answers. I got part (a) a=16, but part (b) f=${x}^{.25}$ I don't understand...
Here is the problem:
This limit represents the derivative of some function f at some number a. State this a and f
$\lim_{{h}\to{0}}$$\frac{\sqrt[4]{16+h}-2}{h}$
Part a) is...
can you please let me know if this sentence is true about optical systems or not?
"Diffraction may limit the resolution achivable by an optical system"
Thanks.
Homework Statement
I want to integrate \int_0^e \ln(x) but first, I wondered if it would be divergent. I figured if xx goes to zero as x goes to zero then the integral would diverge (because xln(x)-x would diverge).
2. The attempt at a solution
I'm wondering how you could show that this limit...
Hello,
sorry I tried to use Latex, but it didn't work...I uploaded picture of what I did instead.
I have a small question about the answer which is infinity why this is positive infinity? I know that correct answer is positive infinity, but I am trying to find explanation why...How do we know...
Hello all,
I have a small question. I was trying to graph this function:
\[\frac{x}{\sqrt{x^{2}+2}}\]
I have calculated it's limit when x goes to infinity, and got 1. I tried the same when it goes to minus infinity, and still got 1, because of the square. The answer should be -1, I don't...
Homework Statement
Find the limit of sin((pi*x*y)/4) as (x, y)--->(-1, 6).
Homework Equations
None.
The Attempt at a Solution
I got 1 as the answer. Am I right?
Homework Statement
Prove that \lim_{x->a}[f(x)+g(x)]=\lim_{x->a}[f(x)]+\lim_{x->a}[g(x)]
Homework Equations
Epsilon/delta definition
The Attempt at a Solution
The book says:
Let the limit of f(x)=L and the limit of g(x)=M. Then,
\mid f(x)-L \mid<\frac{\epsilon}{2} whenever 0<\mid x-a...
The Chandrasekhar Limit is defined as the maximum mass of a white dwarf which is said to be 1.44 solar masses.
My doubt here is if it is defined as being the mass of 1.44 suns then the sun should not even be burning fuel right now. Only then will its mass remain the same.
The Sun keeps burning...
We have the following limit:
\lim _{N\rightarrow \infty}N\log\left(1+\frac{(s\log N)^{2}}{4\pi^{2}} \right )-\sum_{n=1}^{N}\log\left(1+\frac{(s\log n)^{2}}{4\pi^{2}} \right )-N\left(\frac{2\log N}{(\log N)^{2}+\frac{4\pi^{2}}{s^{2}}} \right )
Where is a complex parameter.
any thoughts are...
People have asked this question on this forum before. Yet no one has answered or done background check up.
Reading this has confused my understanding of Special Relativity even more.
http://partners.nytimes.com/library/national/science/053000sci-physics-light.html...
In my textbook it says if you are comparing limn->infinity of an/bn an>0 and bn>0 for the limit comparison test to apply.
It says nothing about "an" having to be greater than "bn", so as long as both are positive for each term I can use the limit comparison test right? It isn't like the...
Hello,
I want to find this limit using L'hospital's rule. I got stuck after doing the derivaties.
\[\lim_{x\rightarrow \infty }\frac{xe^{\frac{x}{2}}}{x+e^{x}}\]
The answer should be 0, can you assist ? Thank you !
The AdS/CFT correspondence is a correspondence of one quantum theory to another quantum theory. But what about the classical limit of these two theories? Is there a correspondence between the corresponding classical theories? If there is, what a precise form this classical-to-classical...
Homework Statement
A process creates a radioactive substance at the rate of 2 g/hr and the substance decays at a rate proportional to its mass, with constant of proportionality k=0.1(hr)^-1. If Q(t) is the mass of the substance at time t, find the limit of Q(t) as t approaches to infinity...
Homework Statement
Use the limit comparison test to prove convergence or divergence for the series sum from n=1 to infinity for ((5n^3)+1)/((2^n)((n^3)+n+1))
Homework Equations
The limit comparison test says that if you have two positive series, sum An and sum Bn, let C=lim n to infinity of...
I have in my notes that $$\lim_{{x}\to{\infty}} \sqrt{x^2 + 2x + 1} - x = 1$$
Is this right? When I calculate it, I get 0, because the square root of infinity is infinity and then I subtract infinity which is 0.
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...
Homework Statement
Use the limit comparison test to show the series converges or diverges: Sum from n=1 to infinity of ((5n^3)+1)/((2^n)((n^3)+n+1))
Homework Equations
suppose Sum An and Sum Bn are two positive series. Let lim as n goes to infinity of An/Bn = c: 1) if 0<c<inifinity then either...
Homework Statement
Use the limit comparison test to check for convergence or divergence: Sum from n=1 to infinity of ((2n)^2+5)^-3
Homework Equations
let lim n to infinity of An/Bn = c
1) if 0<c<infinity then either both converge or both diverge
2) if c=0 and sum Bn converges, so does sum An...
Can someone here help me establish that $\lim\limits_{t \rightarrow \infty} S(t) = \Lambda/\mu$,
given that:
$\frac{dS}{dt}=\Lambda-\mu S-\beta \frac{S}{N}(H+C+C_1+C_2)-\tau \frac{S}{N}(T+C)$
$N(t)=S(t)+H(t)+C(t)+C_1(t)+C_2(t)$
$\Lambda,\beta,\tau > 0$
$\text{As} \ \ t \rightarrow \infty, \...
Ok so I was trying to design a flywheel for a project of mine and an idea occurred to me. Why not use a gas instead of a solid wheel so you didn't haven't worry about it exploding? Could you spin a conductive gas using electromagnets situated around the gas chamber? Whats the maximum speed at...
This is a basic formula but can't find any proof of it. If anyone can explain or give a link showing proof it would be helpful.
## \lim_{x\rightarrow a} f(x)^{g(x)} ## = ##e^{\lim_{x\rightarrow a} g(x)[f(x)-1]}##
What is the proof for it?
This might be a pretty stupid question. But why is it that while applying limits to an exponential function like- \lim_{x\rightarrow 0} e^{f(x)} we move the limit to only the part of the expression which involves the variable on which the limit is being evaluated and hence we now write it as-...
Homework Statement
I ultimately want to discuss convergence of the integral
\int_{0}^{\infty}\frac{1}{\sqrt{x}e^{\sqrt{x}}}dx[/B]Homework Equations
\int_{c}^{\infty}\frac{dx}{x^{p}}
is convergent near x approaching infinity for p>1
3. The Attempt at a Solution
While I understand that the...
Well, this is probably a stupid question, but I don't see why (yet).
Let Xi be random variables identically distributed, with mean 0, such that the cumulative distribution is = 0 for all -1 < x < 1. So, I believe it is clear that for all n, the cumulative distribution of Z = (X1 + X2 ... Xn)/n...
Homework Statement
How can I fount the following limit using L'H'opetal rule?
$$\lim_{x\rightarrow+\infty}(ln(2x)-ln(x-4))$$
Homework EquationsThe Attempt at a Solution
I tried to use the low
$$\lim_{x\rightarrow a} f(x)=\lim_{x\rightarrow a}e^{ln(f(x))}=e^l$$
But it seems to be useless
Many...
Hi,
I'm trying to find the limit as x tends to zero of the function (tanhx-x)/x
this is what i have but i have no idea if i am on the right lines?
lim x-> 0 can be split up into two problems:
limx->0 (tanhx)/x - limx->0 x/x
limx->0 x/x = 1
limx->0 (tanhx)/x can be expressed as
limx->0...
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...
Can anyone tell me how to solve the following limit by factorization method
$\lim{{x}\to{5}} \frac{x^3 + 3x^2 - 6x + 2}{ x^3 + 3x^2 - 3x - 1}$?Please tell me how to factorize such big equation?
Homework Statement
I have a problem, I don't know how to find the limit (cos(Pi/2x))^2x when x is ∞
Homework Equations
(1+(1/x))^(1/x)=e
The Attempt at a Solution
I have been looking for solutions on the internet, but most of these just tend to be for fractions, I don't know how to operate...
Hey! :o
In a space with measure $1$, $||f||_p$ is an increasing function with respect of $p$. To show that $\lim_{p \rightarrow \infty} ||f||_p=||f||_{\infty}$ we have to show that $||f||_{\infty}$ is the supremum, right??
To show that, we assume that $||f||_{\infty}-\epsilon$ is the...
Hello,
Prove that
$$\lim_{{x}\to{0}} \frac{1}{x}$$
Does not exist by contradiction. So the obvious step:Assume:$$\lim_{{x}\to{0}} \frac{1}{x} = M$$
$| 1/x - M| < \epsilon$ for $|x| <\delta_1$
Any ideas? PLEASE DO NOT SOLVE.
Hi, I've been doing limit problems, and just got to this problem and I can't solve it. I would love some tips; you don't have to solve my problem.
Screenshot by Lightshot
Homework Statement
Let a1=0, a2=1, and a(n+2)=n*a(n+1)+an/n+1
a)Calculate the value of a6 and a7
b)Prove that (an) converges.
c)Show that lim an=1-e-1 when n goes to infinity.The Attempt at a Solution
I got the a part and found out that a6 19/30 and a7)91/144
part b)
each subsequent term...
In a proof.
Prove that **given**:
$$\lim_{x \to a} f(x) = L$$ then
$$\lim_{x\to a} |f(x)| = |L|$$
We know that
$$|f(x) - L| < \epsilon \space \text{for} \space |x - a| < \delta_1$$
What is the objective then?
Do we prove there exists a $\delta_2$ such that $\displaystyle \lim_{x\to a}...
I have been wondering about how the universe created itself from nothing and it seems in spacetime, the time dimension must have come first followed by multidimensional space. Following this, matter and dark matter must have been created. While matter and dark matter both underwent timelike...
Homework Statement
Show if this sequence (with n=1 to infinity) diverge or converge
Homework Equations
[/B]
The Attempt at a Solution
If I use the Limit Comparison Test:
compare with so you get that equals lim n -> inf => inf.
Can I use the Test like this? What does this...
Homework Statement
A child is standing at the edge of a straight highway watching her grandparents' car driving away at a constant 22.4 m/s. The air is perfectly clear and steady, and after 8.3 minutes the car's two taillights (654 nm) appear to merge into one. Assuming the diameter of the...