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
For $$\lim _{ x\rightarrow \infty }{ \frac { { x }^{ 2 }+{ e }^{ -{ x }^{ 2 }\sin ^{ 2 }{ x } } }{ \sqrt { { x }^{ 4 }+1 } } } $$, determine whether it exists. If it does, find its value. if it doesn't, explain.
Homework Equations
Sand witch theorem and arithmetic rule...
Homework Statement
Evaluating the following formula: The Attempt at a Solution
Since the integral part is unknown, dividing the case into two: converging and diverging
If converging: the overall value will always be 0
If diverging: ...?
Homework Statement
Hi,
I have been trying to find the limit of the following expression in order to determine the Big-o(in particular, the order of decaying to zero)
\lim_{x\rightarrow 0} \frac {\cos (x) -1 +\frac {x}{2} }{x^4}
Is there another "reasonable" way to solve it?
Thank you...
Homework Statement
1. Lim x→0 Sin(x) * sqrt(1 + 1/x^2) Picture: https://i.gyazo.com/2f61c3c09d32447d4190fbdcd3f2f1e5.png
2. Limx→0 Sin(x)/sqrt(x^2 + x^3) Picture: https://i.gyazo.com/b50081d459ed61bcf1d4ae5baecfa7fa.pngHomework EquationsThe Attempt at a Solution
What I did with the first was...
I regret to say that I know little of engineering, but, to make a long story short, I'm nagged by two questions about energy conversion efficiency. There is a thread i made titled "giant railguns recycling their own energy in space" in the science fiction & fantasy forum here that explains my...
Homework Statement
##\lim_{\alpha\to\omega}-\frac{\alpha r_0}{\omega(\omega^{2}-\alpha^{2})}\sin(\omega t)+\frac{r_0}{\omega^{2}-\alpha^{2}}\sin(\alpha t)##
Homework Equations
I feel I will need to use fact ##\frac{d}{d\omega}\sin(\omega t)=t\cos(\omega t)##
The Attempt at a Solution [/B]...
Homework Statement
Evaluate ##\lim_{a \rightarrow b} \frac{a^b-b^a}{a^a-b^b}##
The attempt at a solution
I applied L'Hospital's rule and to differentiate it, I took the help of logs.
Homework Statement
Calculate: PLIM (probability limit) \frac{1}{T} \sum^T_{t=2} u^2_t Y^2_{t-1}
Homework Equations
Y_t = \rho Y_{t-1} + u_t, t=1,...T, |\rho| <1 which the autoregressive process of order 1
E(u_t) = 0, Var(u_t) = \sigma^2 for t
cov(u_j, u_s) = 0 for j \neq s
The Attempt...
Homework Statement
lim (x)(sin(1/x))
x->∞
Homework EquationsThe Attempt at a Solution
The correct answer is 1, however I do not understand why. I thought that 1/∞ is essentially 0 and sin(0) = 0 so the whole limit would be 0. When I graphed it, the limit did seem to approach 1 though. I...
Homework Statement
For f(x,y) = (2x - y^2)/(2x^2 + y), what is the limit as (x,y)->(0,0)?
Homework EquationsThe Attempt at a Solution
From this image, it seems that the limit would be non-existent since on one side of the sheet, it goes up and up to infinity whereas from the other side, it...
I am designing a custom socket for an application at work and I am concerned about the wall thickness in the area marked by the red box (see attached drawing). The socket will be used to torque a nut to 500Nm.
I have a degree in mechanical engineering and I can do stress calcs on straight...
Homework Statement
Let ##\sum^{\infty}_{n=0} a_n(z-a)^n## be a real or complex power series and set ##\alpha =
\limsup\limits_{n\rightarrow\infty} |a_n|^{\frac{1}{n}}##. If ##\alpha = \infty## then the convergence radius ##R=0##, else ##R## is given by ##R = \frac{1}{\alpha}##, where...
Homework Statement
$$\lim _{x \rightarrow 1} (\frac{23}{1-x^{23}}-\frac{11}{1-x^{11}})$$Homework Equations
i) For functions f and g which are differentiable on an open interval I except possibly at a point c contained in I, if
, and
exists, and
for all x in I with x ≠ c,
then...
Homework Statement
Find tangent line of y=xe^{\frac{1}{x}} at point x=\alpha and it's limit position when \alpha \rightarrow +\infty.
Homework Equations
Tangent of y=f(x) at point M(x_0,f(x_0)): y-y_0=f^{'}(x_0)(x-x_0)
The Attempt at a Solution
Applying the above equation for tangent of...
Homework Statement
Find \lim_{x\to \infty} \sqrt{x^2+1}-x
Homework EquationsThe Attempt at a Solution
This is mostly for a refresher. I know it's zero because when we multiply the top and bottom by the conjugate to obtain \frac{1}{\sqrt{x^2+1}+x} and the denominator increases without bound...
Homework Statement
Find the limit \lim\limits_{t\to -1}\frac{\sqrt[3]{t}+1}{\sqrt[5]{t}+1}. Needless to say: No L'Hopital's rule, otherwise this thread would not exist.
Homework EquationsThe Attempt at a Solution
Have tried multiplying the fraction such that I get the difference of squares in...
Homework Statement
This homework isn't for a course. There is a book on hockey physics with a section discussing puck drop below intended target. When I use the book's formula modified for drag and lift, it seems to show that a 100 mph shot, taken from 30 feet from the net, can not hit the...
If my understanding is correct the definition of a derivative is lim h->0 (f(x+h)-f(x))/h However, I've also seen this used: lim x->c (f(x)-f(x))/(x-c) are these both considered valid definition for the derivative or does the derivative have to tend towards zero? I am a bit confused because I...
Good afternoon
1. Homework Statement
Draw graph of the following equation
Homework Equations
\frac{X-5}{X^2+X-6} = y
The Attempt at a Solution
my problem is searching for the vertical asymptote
from what I know the way to find the vertical asymptote is by limiting the equation near to ∞...
Homework Statement [/b]
Determine if the limit exists as a number, ∞, -∞ or DNE
lim x->4- -(2)/(sqrt(4-x))
The Attempt at a Solution
lim x->4-...
I honestly don't know how to solve. Because I don't know what to do with the sqrt function. If someone could lead me in the right direction here...
Show that $\displaystyle\lim_{p\to +\infty}C_p ((0,0); 1)=C_{\infty}((0,0);1).$
Hello, I think that this limit is infinite. At $C_p=\{(x,y)\in \mathbb{R}^2: |x|^p+|y|^p=1, \, p>1\}$, then is reasonably think. But how can show this?
Homework Statement
prove: \lim_{n\rightarrow \infty} {\frac{n!}{2^n}}=\infty
Homework Equations
Def. of a limit
The Attempt at a Solution
I would like to know if my solution is right or not. I think it is right but I would like to get a feedback. Please do not give me the answer, just...
Homework Statement
Calculate the following limit if it exists
## lim_{z\to -1}\frac{\sqrt{z}-i+\sqrt{z+1}}{\sqrt{z^2-1}} ##
the branch of root is chosen to that ##\sqrt{-1}=i##
Homework Equations
3. The Attempt at a Solution
[/B]
By inserting ##z=-1## directly, I get a ##\frac{0}{0}##...
Homework Statement
Show that ##\lim _{ n\rightarrow \infty }{ \left( \frac { \sqrt { n+c } +d }{ \sqrt [ 3 ]{ { n }^{ 2 }+an+b } } \right) } =0,\quad n>-c ##
Homework Equations
Sandwich theorem
The Attempt at a Solution
Ok, So I know my method is extremely long, I'm just wandering if 1)...
Consider a ##j## point all massive leg one loop polygonal Feynman diagram ##P## representing some scattering process cut on a particular mass channel ##s_i##. Invoking the relevant Feynman rules and proceeding with the integration via dimensional regularisation for example gives me an expression...
Hi,
My instructor gave us a challenge problem to solve in limits: ##\lim _{ x\rightarrow \infty }{ \left[ { x }^{ 2 }\left( 1-\cos { \frac { 5.1 }{ x } } \right) \right] } ## Note that we did not take Hospital's rule yet so we couldn't have used it.
Now my first thought was to use the...
Hi. I am self-studying GR and have many questions. Here are a few. If anyone can help me with any of them I would be grateful.
1 - What is the difference between Tu v and Tvu ?
2 - I have read that the order of indices matters in tensors but when transforming tensors from one coordinate...
Hello
Why does Mineral Insulated heating cables have such limited power output per feet and maximum operating voltages?
I thought it has to do with the dielectric strength of the MgO, that is used as an insulation in the coaxial cable, but the dielectric strength for MgO is higher than the...
Hi all! In the midst of making up limit problems to solve generally, I came across a limit in which I'm not sure of the answer. This is not homework, but merely a product of free time.
Homework Statement
Find the limit.
## lim_{x \to a^+} {\frac{\sqrt(a^2 - x^2)}{a}} , a > 0 ##
The attempt...
I have done this problem before but forgot how to get from one step to the next:
let a>0.
how is absval(x^1/2-a^1/2) equal to abval(x-a)/(x^1/2-a^1/2)?
Homework Statement
Given f(x,y)=(y+x)/(y-x) use an ε-∂ proof to show that lim(x,y)→(0,1) f(x,y) exists.
Homework Equations
|(y+x)/(y-x)-1|=|(2x)/(y-x)|
The Attempt at a Solution
I know that the limit is 1. I can't figure out how to massage the above any further to get it into the form...
To find the horizontal asymptotes of a rational function, we find the limit as x goes to infinity. Given the rational function ##\displaystyle\frac{x + 1}{\sqrt{x^2+1}}##, we can find the limit by multiplying the numerator and the denominator by ##\frac{1}{x}##. This gives us ##\frac{1 +...
Hi guys, I don´t understand too much the Newtonian limit of General Relativity. My question is:
In this limit ga b (x) = ηa b (x) + ha b (x) where O(h2→0). Then, according to GR, it's straightforward to demonstrate that the Einstein equation Gt t = k*Tt t in that limit leads to Poisson equation...
Dear PF Forum,
In previous threads, I have asked about sine and cosine. The answer given by the members/mentors/advisor are very clear. But lengthy. Perhaps these yes/no questions that I can simply remember and not forget it (again).
So here we are
1. if h = 0 then sin(h) = 0
2. if ##\lim_{h...
Homework Statement
The limit
##\lim_{x\to\pi}\frac{xcos\frac{x}{2}}{\pi^{2}-x^{2}}##
Can be expressed as a fraction. Solve
2. Relevant equation
3. The Attempt at a Solution
EDIT
See new post for solution
Dear PF Forum,
Continuing our debate discussion in differential in slice of X.
I read this particulare website. About proofing the derivative of sine(x).
http://tutorial.math.lamar.edu/Classes/CalcI/ProofTrigDeriv.aspx
In there, the web writes
arc AC < |AB| + |BC|
< |AB| + |BD|...
I've been trying to figure this out for days. I'm told that atmospheric pressure imposes a limit on maximum possible exhaust velocity in the Earth's atmosphere, and that under STP conditions that limit is approximately 15,000 feet per second. But that doesn't make any sense. Suppose you had...
I've read that matter and energy fields occupy all of space, and that space is integral with time, so I'm wondering whether it might be consistent with contemporary physics for undetectably small proportions of the mass of matter and / or energy to occupy arbitrarily large proportions of an...
Hi,
This is a question regarding Example 3.6 in Section 3.5 (p.35) of 'QFT for the Gifted Amateur' by Lancaster & Blundell.
Given, [a^{\dagger}_\textbf{p}, a_\textbf{p'}] = \delta^{(3)}(\textbf{p} - \textbf{p'}) . This I understand. The operators create/destroy particles in the momentum state...
$y_0=k$ where $k$ is a constant.
$x_{n+1}=30-\dfrac{y_n}{2}$
$y_{n+1}=30-\dfrac{x_{n+1}}{2}$
Prove that $(x_n, y_n)$ converges to $(20, 20)$ for all values of $k$.
My attempt:
I wrote a computer program and verified this for a few values of $k$. But I don't know how to prove that $x_n$ and...
Hello,Does anyone know any websites that offer limits exercise? I googled it and didn't find much, I just want limits and nothing else. Also, do limits generators exist?
Homework Statement
Let ##x_n## be the solution to the equation
##\left( 1+\frac{1}{n} \right)^{n+x} = e##
Calculate ##\lim_{n\to \infty} x_n##
Homework Equations
N/A
The Attempt at a Solution
Since ##\lim_{n \to \infty} \left(1+ \frac{1}{n} \right) = e## that tells me that ##\lim_{n\to...